New Graph

 [7, 7, 7, 6, A, A, B, C, B, C, 1, 5], [6, 8, 8, 7, 3, 3, A, B, C, 2, 4, 9]

 

 

π = [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2]

POSSIBLE RANKS

1 x 16
2 x 8
4 x 4

BASE DETERMINANT 3762658697/34359738368, .1095077808

NullSpace of Δ

{1, 4}, {2, 3, 10, 11, 12}, {5, 9}, {6, 7, 8}

 
1 . Coloring, {}

Ωp(Δ)=0:     p = s 2 - 3s 3 - 6s 4 - 8s 5 - 24s 6 - 32s 7 - 32s 8

R: [7, 7, 7, 6, A, A, B, C, B, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 4 vs 7 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y2, 0, 0, 0, y1, -y1 + y3, y4, 0, 0, y2, y3, y4]

  p' = s 3 - s 6   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, 3 y1, -3 y1 - 3 y3 + 5 y5 - 3 y7 + 8 y8, -3 y2 - 3 y4 + 8 y5 - 3 y6 + 5 y8, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p' = s 3 - s 9   p = s 3 - s 9


M              \ ;       N

$ [ [0, 0, 0, 0, 384, 60, 0, 0, 0, 332, 0, 428] , [0, 0, 0, 308, 0, 166, 0, 286, 0, 223, 0, 221] , [0, 0, 0, 332, 0, 222, 0, 194, 0, 349, 0, 107] , [0, 308, 332, 0, 0, 0, 60, 0, 384, 0, 120, 0] , [384, 0, 0, 0, 0, 0, 378, 0, 0, 0, 442, 0] , [60, 166, 222, 0, 0, 0, 320, 0, 214, 0, 222, 0] , [0, 0, 0, 60, 378, 320, 0, 320, 0, 446, 0, 884] , [0, 286, 194, 0, 0, 0, 320, 0, 164, 0, 240, 0] , [0, 0, 0, 384, 0, 214, 0, 164, 0, 442, 0, 0] , [332, 223, 349, 0, 0, 0, 446, 0, 442, 0, 616, 0] , [0, 0, 0, 120, 442, 222, 0, 240, 0, 616, 0, 768] , [428, 221, 107, 0, 0, 0, 884, 0, 0, 0, 768, 0] ] $     $ [ [0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1] , [0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1] , [0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0] , [1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0] , [1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0] , [0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0] , [0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0] , [0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0] ] $

τ= 72 , r'= 1/2

R: [7, 7, 7, 6, A, A, B, C, B, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, B, C, 2, 4, 9]

Ranges

Action of R on ranges, [[24], [24], [25], [18], [20], [24], [25], [25], [18], [20], [24], [25], [25], [18], [22], [22], [2], [29], [3], [29], [29], [3], [30], [30], [19], [30], [4], [30], [4], [1]]
Action of B on ranges, [[11], [11], [6], [21], [23], [12], [27], [7], [26], [23], [12], [27], [7], [26], [24], [25], [15], [13], [10], [13], [14], [10], [29], [8], [28], [30], [17], [9], [5], [16]]

Cycles:    R , {{1, 7, 11}, {5, 10, 12}},   B , {{9, 12}, {2, 4, 7, 8, 10, 11}}

β({1, 5}) = 12/301
β({1, 6}) = 15/2408
β({1, 10}) = 83/2408
β({1, 12}) = 107/2408
β({2, 4}) = 11/344
β({2, 6}) = 83/4816
β({2, 8}) = 143/4816
β({2, 10}) = 223/9632
β({2, 12}) = 221/9632
β({3, 4}) = 83/2408
β({3, 6}) = 111/4816
β({3, 8}) = 97/4816
β({3, 10}) = 349/9632
β({3, 12}) = 107/9632
β({4, 7}) = 15/2408
β({4, 9}) = 12/301
β({4, 11}) = 15/1204
β({5, 7}) = 27/688
β({5, 11}) = 221/4816
β({6, 7}) = 10/301
β({6, 9}) = 107/4816
β({6, 11}) = 111/4816
β({7, 8}) = 10/301
β({7, 10}) = 223/4816
β({7, 12}) = 221/2408
β({8, 9}) = 41/2408
β({8, 11}) = 15/602
β({9, 10}) = 221/4816
β({10, 11}) = 11/172
β({11, 12}) = 24/301

Partitions
α([{1, 2, 3, 7, 9, 11}, {4, 5, 6, 8, 10, 12}]) = 1/1

b1 = {1, 2, 3, 7, 9, 11} ` , ` b2 = {4, 5, 6, 8, 10, 12}

Action of R and B on the blocks of the partitions: = [1, 2] [2, 1]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {}
Rank2
R,B [7, 7, 7, 6, A, A, B, C, B, C, 1, 5], [6, 8, 8, 7, 3, 3, A, B, C, 2, 4, 9]
π2 [0, 0, 0, 384, 60, 0, 0, 0, 332, 0, 428, 0, 308, 0, 166, 0, 286, 0, 223, 0, 221, 332, 0, 222, 0, 194, 0, 349, 0, 107, 0, 0, 60, 0, 384, 0, 120, 0, 0, 378, 0, 0, 0, 442, 0, 320, 0, 214, 0, 222, 0, 320, 0, 446, 0, 884, 164, 0, 240, 0, 442, 0, 0, 616, 0, 768]
u2 [0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1] (dim 1)
wpp [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]

 

 
2 . Coloring, {2}

R: [7, 8, 7, 6, A, A, B, C, B, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[3 y3, 0, 0, 0, 3 y2, -3 y2 + 3 y4, 3 y1, -3 y2 + 3 y4, 0, 3 y2, -3 y3 + 7 y4 - 3 y1, 3 y4]

  p' = - s 3 + s 6   p' = - s 4 + s 7   p = s 2 - s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y7 - 3 y6 - 3 y5 - 3 y4 + 13 y3 - 3 y2 - 3 y8 + 13 y9, 3 y1, 3 y7, 0, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2, 3 y8, 3 y9]

  p = - s 6 - s 7 + s 9 + s 10


 
3 . Coloring, {3}

R: [7, 7, 8, 6, A, A, B, C, B, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-3 y1 + 7 y2 - 3 y4, 0, 0, 0, 3 y3, -3 y3 + 3 y2, 3 y1, -3 y3 + 3 y2, 0, 3 y3, 3 y4, 3 y2]

  p = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y3 + 5 y5 - 3 y7 + 8 y8, -3 y2 - 3 y4 + 8 y5 - 3 y6 + 5 y8, 3 y2, 0, 3 y1, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 3 + s 9   p' = - s 3 + s 9


 
4 . Coloring, {4}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 6 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[16 y1 + 7 y3 - 27 y2, 0, 0, 0, 4 y1 + 3 y3 - 7 y2, 0, 2 y1, 0, 0, 10 y1 + 4 y3 - 16 y2, 2 y3, 2 y2]

  p' = s 2 - s 5   p' = s - s 4   p = - s + s 4

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y3 - 3 y4 - 3 y2 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y3, 0, 3 y4, 0, 3 y2, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
5 . Coloring, {5}

Ωp(Δ)=0:     p = 6s 2 - s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, C, 1, 5]
B: [6, 8, 8, 7, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, 0, y1, 0, y2, y3, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -2 y3 + 11 y2 + 11 y1 - 2 y6 + 11 y5 - 39 y4 + 11 y7 - 2 y8, 2 y3, 2 y2, 0, 2 y1, 2 y6, 2 y5, 2 y4, 2 y7, 2 y8, 3 y2 + 3 y1 + 3 y5 - 11 y4 + 3 y7]

  p' = - s 3 + s 9   p = - s 3 + s 9


 
6 . Coloring, {6}

Ωp(Δ)=0:     p = - 2s 2 - 5s 3 + 2s 4 + 16s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, C, 1, 5]
B: [6, 8, 8, 7, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y2, 0, y1, 0, -2 y2 - 2 y1 + 5 y3 + 5 y4 - 4 y5, y3, 4 y2 + 4 y1 - 6 y3 - 6 y4 + 5 y5, 0, 0, 5 y2 + 5 y1 - 8 y3 - 8 y4 + 6 y5, y4, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -6 y1 - 3 y5 + 5 y4 - 3 y3 + 8 y7, 3 y1, -3 y6 + 8 y4 - 3 y2 + 5 y7, 0, 3 y1, 3 y5, 3 y6, 3 y4, 3 y2, 3 y3, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8   p' = - s 3 + s 9


 
7 . Coloring, {7}

R: [7, 7, 7, 6, A, A, A, C, B, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, y5, y4, y3, 0, 0, y2, y4, y1]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 6 y1, -3 y3 - 9 y1 - 3 y2 - 3 y4 + 13 y5 - 3 y6 + 13 y7, 3 y3, 0, 3 y1, 3 y2, 3 y4, 3 y5, 0, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9


 
8 . Coloring, {8}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 4 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 4 y1 - 2 y2 - 7 y3 + 3 y4, 2 y2, 16 y1 - 27 y3 + 7 y4, 0, 0, 2 y3, 2 y4, 10 y1 - 16 y3 + 4 y4 ]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y4, 2 y2, 0, y2, 2 y4 - 3 y2, y3, y6, y7, 0, y5]

  p' = - s 6 + s 8   p = s 6 - s 8

 ยป SYNC'D !RANK'D


 
9 . Coloring, {9}

Ωp(Δ)=0:     p = - 9s 3 - 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, -27 y1 - 2 y2 + 16 y4 + 7 y3, 2 y2, -7 y1 + 4 y4 + 3 y3, 0, 0, 2 y4, -16 y1 + 10 y4 + 4 y3, 2 y3 ]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y1, y2, 0, y8, y7, y6, 2 y8, y5, y4, 0]

  p = s 3 - s 9


 
10 . Coloring, {10}

R: [7, 7, 7, 6, A, A, B, C, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y7, 0, 0, y2, y6, y3, 0, 0, y4, y5, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y4, 0, y2, y3, y5, y6, y9, y7, y8]


 
11 . Coloring, {11}

R: [7, 7, 7, 6, A, A, B, C, B, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y5 - 3 y3 - 3 y4 + 13 y6 - 3 y7 - 3 y8 + 13 y9, 3 y1, 3 y2, 0, 0, 3 y5, 3 y3, 3 y4, 3 y6, 3 y7, 3 y8, 3 y9]

  p = - s 4 - s 5 + s 9 + s 10


 
12 . Coloring, {12}

R: [7, 7, 7, 6, A, A, B, C, B, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, 0, 0, 0, 0, y1, y2, 0, y3, y6, y5, y7]

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, y9, y5, y6, 0, y7, y8, y9]

  p = - s 4 + s 10


 
13 . Coloring, {2, 3}

R: [7, 8, 8, 6, A, A, B, C, B, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[-18 y1 + 14 y2 + 9 y4 + 2 y3, 0, 0, 0, y1, -5 y1 + 4 y2 + 2 y4 + y3, y2, -10 y1 + 8 y2 + 4 y4 + 2 y3, 0, y4, y3, -16 y1 + 13 y2 + 8 y4 + 2 y3]

  p' = - s 3 + s 6   p = - s 2 + s 8   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -6 y1 - 6 y5 - 3 y2 + 13 y3 - 3 y4 + 13 y6, 3 y1, 3 y1, 0, 3 y5, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p' = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 9   p = - s 3 - s 4 + s 6 + s 7


 
14 . Coloring, {2, 4}

R: [7, 8, 7, 7, A, A, B, C, B, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y2, 0, 0, 0, y2, 0, y1 + y3, y1, 0, y3, y4, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, -3 y7 + 5 y5 - 3 y4 + 5 y2, 3 y1, -3 y1 - 3 y8 - 3 y6 - 3 y3 + 8 y5 + 8 y2, 0, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2]

  p = s - 2s 3 - 3s 4 - 2s 5 - s 6 + 2s 8 + 3s 9 + 2s 10   p = - s - s 2 + s 4 + s 5 + s 6 + s 7 - s 9 - s 10


 
15 . Coloring, {2, 5}

R: [7, 8, 7, 6, 3, A, B, C, B, C, 1, 5]
B: [6, 7, 8, 7, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, 0, y7, 0, y6, y5, y4, y5, 0, y3, y2, y1]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y9 - 3 y4 - 3 y5 - 3 y6 + 13 y7 - 3 y8 - 3 y3 + 13 y2, 3 y1, 3 y9, 0, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 3 y3, 3 y2]

  p = - s 6 - s 7 + s 9 + s 10


 
16 . Coloring, {2, 6}

R: [7, 8, 7, 6, A, 3, B, C, B, C, 1, 5]
B: [6, 7, 8, 7, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[-2 y1 - 20 y3 + 16 y2 - 27 y4 + 7 y5, 0, 2 y1, 0, -12 y3 + 10 y2 - 16 y4 + 4 y5, 2 y3, 2 y2, 2 y3, 0, 2 y4, 2 y5, -4 y3 + 4 y2 - 7 y4 + 3 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -6 y2 - 3 y1 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y2, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9   p = s 5 - s 7 - s 8 + s 10


 
17 . Coloring, {2, 7}

R: [7, 8, 7, 6, A, A, A, C, B, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, y1, y5, y2, y5, 0, y3, y5, y4]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 6 y2, -3 y1 - 9 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 + 13 y7, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 0, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9


 
18 . Coloring, {2, 8}

R: [7, 8, 7, 6, A, A, B, B, B, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 4 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[-27 y4 - 20 y3 + 7 y2 + 16 y1, 0, 0, 0, 2 y4, 2 y3, 2 y2, 2 y3, 0, -16 y4 - 12 y3 + 4 y2 + 10 y1, 2 y1, -7 y4 - 4 y3 + 3 y2 + 4 y1]

  p = - s 2 + s 8   p' = - s 3 + s 6   p = - s 2 + s 5   p' = s 2 - s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y5, y4, 2 y3, 0, y3, y2, y1, y4 - y3 - y1 + y6, -y5 - y2 + 2 y4 - 2 y3 + 2 y6, 0, y6]

  p' = s 4 - s 7   p = s 4 - s 7   p' = s 5 - s 8


 
19 . Coloring, {2, 9}

R: [7, 8, 7, 6, A, A, B, C, C, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y3, 0, 0, 0, 5 y3 - 2 y1 - y2 - y4, y1, y3, y1, 0, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y4, y7, y8, 0]

  p = - s 6 + s 9


 
20 . Coloring, {2, 10}

R: [7, 8, 7, 6, A, A, B, C, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[7 y1, 9 y1 - 7 y2 - 7 y7 + 9 y6 - 7 y5 - 7 y4 + 9 y3 - 7 y8, 0, 0, 7 y2, 7 y7, 7 y6, 7 y5, 0, 7 y4, 7 y3, 7 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1 - y2 + y8 - y7 - y6 - y5 + y4 + y3, 0, y2, y8, y7, y6, y5, y4, y3]

  p = - s 8 + s 9


 
21 . Coloring, {2, 11}

R: [7, 8, 7, 6, A, A, B, C, B, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, 2 y4, y4, 0, y5, y7, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[7 y3 - 2 y1 - 2 y2 - 2 y4 - 2 y5, 2 y3, 2 y1, 0, 0, 2 y2, 2 y3, 2 y4, 3 y3 - 2 y6, 2 y3, 2 y5, 2 y6]

  p = - s - s 2 + s 6 + s 7   p = s - s 3 - s 6 + s 8   p = - s - s 4 + s 6 + s 9   p = s - s 5 - s 6 + s 10


 
22 . Coloring, {2, 12}

R: [7, 8, 7, 6, A, A, B, C, B, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, y9, y5, y6, 0, y7, y8, y9]

  p = - s 7 + s 10


 
23 . Coloring, {3, 4}

R: [7, 7, 8, 7, A, A, B, C, B, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y4, 0, 0, 0, y4, 0, y1 + y2, y1, 0, y2, y3, y3]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, -3 y1 - 3 y2 + 5 y4 - 3 y6 + 8 y7, 3 y8, -3 y8 - 3 y3 + 8 y4 - 3 y5 + 5 y7, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p' = - s + s 9   p = - s + s 9


 
24 . Coloring, {3, 5}

R: [7, 7, 8, 6, 3, A, B, C, B, C, 1, 5]
B: [6, 8, 7, 7, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[7 y1, 0, 9 y1 - 7 y7 - 7 y8 + 9 y5 - 7 y6 - 7 y2 + 9 y3 - 7 y4, 0, 7 y7, 7 y8, 7 y5, 7 y6, 0, 7 y2, 7 y3, 7 y4]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y4 - 3 y3 + 5 y5 - 3 y7 + 8 y6, -3 y1 - 3 y2 + 8 y5 - 3 y8 + 5 y6, 3 y1, 0, 3 y4, 3 y3, 3 y2, 3 y5, 3 y8, 3 y7, 3 y6]

  p = - s 3 + s 9   p' = s 3 - s 9


 
25 . Coloring, {3, 6}

R: [7, 7, 8, 6, A, 3, B, C, B, C, 1, 5]
B: [6, 8, 7, 7, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[4 y5 - 16 y4 + 10 y3 + 10 y2 + 4 y1, 0, -2 y6 + 7 y5 - 27 y4 + 16 y3 + 16 y2 + 7 y1, 0, 2 y6, 2 y5, 2 y4, 2 y3, 0, 2 y2, 3 y5 - 7 y4 + 4 y3 + 4 y2 + 3 y1, 2 y1 ]

  p = s 4 - s 7   p' = s 4 - s 7   p' = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y7 - 3 y6 + 5 y4 - 3 y2 + 8 y1, 3 y7, -3 y7 - 3 y5 + 8 y4 - 3 y3 + 5 y1, 0, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2, 3 y1]

  p' = - s 2 + s 8   p = - s 2 + s 8   p' = - s 3 + s 9


 
26 . Coloring, {3, 7}

R: [7, 7, 8, 6, A, A, A, C, B, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, y5, y4, y3, y4, 0, y2, y4, y1]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 6 y4, 3 y1, -9 y1 - 3 y5 + 13 y6 - 3 y2 + 13 y3, 0, 3 y4, 3 y5, 6 y1 - 9 y4, 3 y6, 0, 3 y2, 3 y3]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8   p = - s 3 + s 9


 
27 . Coloring, {3, 8}

R: [7, 7, 8, 6, A, A, B, B, B, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 10 y2 + 8 y3 - 16 y4 + 4 y1, 2 y3, 2 y2, 2 y3, 0, 3 y1 + 4 y2 + 4 y3 - 7 y4, 16 y2 + 16 y3 - 27 y4 + 7 y1, 2 y4]

  p = - s 2 + s 5   p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1, y2, 2 y3, 0, y3, y4, y5, y7, y6, 0, y8]

  p = - s 7 + s 9


 
28 . Coloring, {3, 9}

R: [7, 7, 8, 6, A, A, B, C, C, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y3, 0, 0, 0, 5 y3 - 2 y1 - y2 - y4, y1, y3, y1, 0, y2, y3, y4]

  p' = - s 3 + s 6   p = - s 2 + s 5   p = - s 2 + s 8   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y4, y7, y8, 0]

  p = - s 3 + s 9


 
29 . Coloring, {3, 10}

R: [7, 7, 8, 6, A, A, B, C, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, y8, 0, 0, y6, y5, y4, y5, 0, y1, y2, y3]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y3 + y7, y1, 0, y3, y2, y3, y4, y5, y7, y6]

  p = - s 6 + s 8   p' = - s 6 + s 8


 
30 . Coloring, {3, 11}

R: [7, 7, 8, 6, A, A, B, C, B, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y4, 2 y3, y3, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[-3 y1 - 3 y4 + 8 y5 - 3 y6 + 5 y8, -3 y2 - 3 y3 + 5 y5 - 3 y7 + 8 y8, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s + s 9   p' = - s + s 9


 
31 . Coloring, {3, 12}

R: [7, 7, 8, 6, A, A, B, C, B, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y6, y7, y6, y3, y4, y5, y2]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y2 - y3 - y4 + y5 + y6 - y7, y1, y2, y3, y8, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9   p = - s 4 + s 10


 
32 . Coloring, {4, 5}

Ωp(Δ)=0:     p = 3s 2 + 2s 3 - 8s 5 + 32s 7   p' = 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, C, B, C, 1, 5]
B: [6, 8, 8, 6, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y6, 0, y2, 0, y3, 0, 0, y4, y5, y7]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y5 - 3 y6 + 13 y7 - 3 y8 - 3 y3 + 13 y4, 3 y1, 3 y2, 0, 3 y5, 0, 3 y6, 3 y7, 3 y8, 3 y3, 3 y4]

  p = - s 3 - s 4 + s 8 + s 9


 
33 . Coloring, {4, 6}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, B, C, B, C, 1, 5]
B: [6, 8, 8, 6, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y3, 0, y1, 0, y2, 0, 4 y3 + 4 y1 + 5 y2 - 6 y4, 0, 0, -2 y3 - 2 y1 - 4 y2 + 5 y4, y4, 5 y3 + 5 y1 + 6 y2 - 8 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 + 5 y4 - 3 y6 + 8 y7, 3 y1, -3 y3 + 8 y4 - 3 y5 + 5 y7, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
34 . Coloring, {4, 7}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, 0, 0, y4, y5, y6]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 + 13 y7, 3 y1, 3 y2, 0, 3 y3, 0, 3 y4, 3 y5, 0, 3 y6, 3 y7]

  p = - s 2 - s 3 + s 7 + s 8


 
35 . Coloring, {4, 8}

Ωp(Δ)=0:     p' = s 2 - 16s 4 - 8s 5 + 16s 6 + 64s 7   p' = s 3 + 4s 4 - 8s 6 - 16s 7   p = s 2 + 56s 5 + 16s 6 - 64s 7 - 256s 8

R: [7, 7, 7, 7, A, A, B, B, B, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 3 vs 6 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y3, 0, 0, 0, y2, 0, y1, 0, 0, y2, -y3 + 5 y2 - y1, y2]

  p' = s - s 4   p = s - s 4   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y5 + y2 - y4, y3, 0, y1, 0, y5, y2, y3, 0, y4]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8


 
36 . Coloring, {4, 9}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7   p = 3s 2 - 10s 3 - 40s 5 + 32s 6 - 32s 7 + 128s 8

R: [7, 7, 7, 7, A, A, B, C, C, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 3 vs 6 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, 0, 0, y1, y2, y3]

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, 0, y5, y6, y6, y7, 0]

  p = - s 3 + s 8


 
37 . Coloring, {4, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -7 y1 - 7 y2 - 7 y6 + 9 y7 + 9 y3 - 7 y4 + 9 y5, 7 y1, 0, 7 y2, 0, 7 y6, 7 y7, 7 y3, 7 y4, 7 y5]

  p = - s 2 - s 3 + s 7 + s 8


 
38 . Coloring, {4, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 6 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 16 y1 - 27 y2 + 7 y3, 4 y1 + 3 y3 - 7 y2, 0, 2 y1, 0, 0, 10 y1 - 16 y2 + 4 y3, 2 y3, 2 y2]

  p' = s 2 - s 5   p' = s - s 4   p = s - s 4

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y6 + 13 y7 - 3 y4 - 3 y5 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 0, 3 y6, 3 y7, 3 y4, 3 y5, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
39 . Coloring, {4, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, 0, y3, y6, y4, y5]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y3 - 2 y7, y1, y2, y3, y4, 0, y5, 0, 2 y7, y6, y7]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
40 . Coloring, {5, 6}

R: [7, 7, 7, 6, 3, 3, B, C, B, C, 1, 5]
B: [6, 8, 8, 7, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, 0, y3, 0, y2, y1, y5, 0, 0, 0, y6, 3 y1]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, 11 y1 - 2 y2 - 2 y3 + 11 y4 - 39 y5 + 11 y6 - 2 y7, 0, 2 y1, 0, 2 y2, 2 y3, 2 y4, 2 y5, 2 y6, 2 y7, 3 y1 + 3 y4 - 11 y5 + 3 y6]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
41 . Coloring, {5, 7}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, C, 1, 5]
B: [6, 8, 8, 7, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y6, y4, 0, 0, y5, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y2 + 3 y6, 3 y2, 3 y3, 0, 3 y6, -6 y2 - 9 y6 - 3 y3 - 3 y1 + 13 y5 - 3 y7 + 13 y4, 3 y1, 3 y5, 3 y6, 3 y7, 3 y4]

  p' = s 4 + s 5 - s 7 - s 8   p = - s 4 - s 5 + s 7 + s 8   p = - s 4 + s 10


 
42 . Coloring, {5, 8}

R: [7, 7, 7, 6, 3, A, B, B, B, C, 1, 5]
B: [6, 8, 8, 7, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y2, 4 y2 - 2 y3, 0, 2 y2 - y3, y3, y6, y4, y5, 0, y7]

  p = - s 6 + s 8   p' = - s 6 + s 8


 
43 . Coloring, {5, 9}

R: [7, 7, 7, 6, 3, A, B, C, C, C, 1, 5]
B: [6, 8, 8, 7, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y4, y8, y7, 0]

  p = - s 3 + s 9


 
44 . Coloring, {5, 10}

Ωp(Δ)=0:     p = - 6s 2 + 5s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y5, y4, 0, -y5 + y4 + 3 y6, y6, y3, 0, 0, -y5 + y4 + 2 y6, y2, y6]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
45 . Coloring, {5, 11}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, C, 4, 5]
B: [6, 8, 8, 7, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y2 - y3 + y4 + y5 - y6 - y7, y1, y2, y3, y4, 0, 0, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 - 3 y8 + 13 y9, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 3 y9]

  p = - s 4 - s 5 + s 9 + s 10


 
46 . Coloring, {5, 12}

Ωp(Δ)=0:     p = 9s 3 + 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, C, 1, 9]
B: [6, 8, 8, 7, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y2, y1, y2 + y8, y8, y7, y6, 0, y5, y4, y8]

  p' = - s 3 + s 9   p = - s 3 + s 9


 
47 . Coloring, {6, 7}

Ωp(Δ)=0:     p = - 2s 2 + 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, C, 1, 5]
B: [6, 8, 8, 7, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1 + y6, 0, y1, 0, y2, y6, y3, 0, 0, y4, y6, y5]

  p' = s 4 - s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1, 3 y2, -3 y1 - 6 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y7 - 3 y8 + 13 y6, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y7, 3 y8, 3 y6]

  p' = s 5 + s 6 - s 8 - s 9   p = s 5 - s 7 - s 8 + s 10

 ยป SYNC'D !RANK'D


 
48 . Coloring, {6, 8}

R: [7, 7, 7, 6, A, 3, B, B, B, C, 1, 5]
B: [6, 8, 8, 7, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-3 y1 - 7 y2 + 14 y3 + 8 y5, 0, 3 y1, 0, -5 y2 + 7 y3 + 7 y5, -7 y2 + 8 y3 - 3 y4 + 14 y5, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y2, 2 y2, 0, y2, y7, y3, y4, y5, 0, y6]

  p' = s 6 - s 8   p = s 6 - s 8


 
49 . Coloring, {6, 9}

R: [7, 7, 7, 6, A, 3, B, C, C, C, 1, 5]
B: [6, 8, 8, 7, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-3 y1 + 3 y2, 0, 3 y1, 0, 7 y2 - 3 y5 - 3 y4, 3 y2 - 3 y3, 3 y2, 0, 0, 3 y5, 3 y3, 3 y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y4, y3, 0, y4, y2, y6, 2 y4, y5, y7, 0]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
50 . Coloring, {6, 10}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y4, y5, 0, y5 + y7, y7, y2, 0, 0, y3, y6, y7]

  p' = - s 5 + s 8   p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y2, y1, 0, y2, y3, y4, y5, y8, y6, y7]

  p = - s 7 + s 9


 
51 . Coloring, {6, 11}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, C, 4, 5]
B: [6, 8, 8, 7, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, -3 y3 + 5 y4 - 3 y5 - 3 y6 + 5 y1 - 3 y2 + 5 y7, 3 y3, 3 y4, 3 y5, 3 y6, 0, 0, 3 y1, 3 y2, 3 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y3 + 8 y4 - 3 y5 + 5 y7, -6 y2 - 3 y1 + 5 y4 - 3 y6 + 8 y7, 3 y2, 0, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8   p' = - s 3 + s 9


 
52 . Coloring, {6, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, C, 1, 9]
B: [6, 8, 8, 7, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, y4, y3, 0, y2, y4, y1, y5 + 2 y4]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, y9, y5, y6, 0, y7, y8, y9]

  p = - s 4 + s 10

 ยป SYNC'D !RANK'D


 
53 . Coloring, {7, 8}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, 0, 0, y5, 2 y3, y6]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 10 y2, 11 y1 - 15 y2 + 11 y4 - 5 y3 - 5 y5 + 11 y6 - 5 y7, 5 y1, 0, 5 y2, 5 y4, 5 y3, 5 y5, 0, 5 y6, 5 y7]

  p = - s 4 + s 6 + s 7 - s 9   p' = - s 4 - s 5 + s 7 + s 8


 
54 . Coloring, {7, 9}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 6 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, y3, y2, 0, 0, y5, 0, y4]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y3, y1, y2, 0, y3, y4, y5, 2 y3, 0, y6, 0]

  p' = s 4 - s 7   p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
55 . Coloring, {7, 10}

R: [7, 7, 7, 6, A, A, A, C, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y1, y2, 0, 0, y1, y5, y3, 0, 0, y4, y5, y5]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -5 y1 - 5 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 5 y1, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7]

  p = s 4 + s 5 - s 7 - s 8


 
56 . Coloring, {7, 11}

R: [7, 7, 7, 6, A, A, A, C, B, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, 3 y5, 0, 0, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y5 - 9 y6 - 3 y2 + 13 y3 - 3 y4 + 13 y7, 6 y6, 3 y1, 0, 0, 3 y5, 3 y6, 3 y2, 3 y3, 0, 3 y4, 3 y7]

  p' = - s 2 - s 3 + s 7 + s 8   p = - s 2 - s 3 + s 7 + s 8


 
57 . Coloring, {7, 12}

R: [7, 7, 7, 6, A, A, A, C, B, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 + y2 - y3 - y4 + y5 + y6, 0, 0, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y7, y1, y2, y3, y7, y4, y5, 0, 0, y6, y7]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
58 . Coloring, {8, 9}

R: [7, 7, 7, 6, A, A, B, B, C, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y4, 0, 0, 0, y3, y2, y1, 0, 0, y4, y3 + y2, y1]

  p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y4, y3, 0, y2, y8, y7, y6, y5, y10, y9]


 
59 . Coloring, {8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y3, y3, y4, 0, 0, y5, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y2, 0, y2, 2 y1 - 3 y2, y3, y5, -3 y1 + 4 y2 + 2 y3, 0, y4]

  p = - s 4 + s 6   p' = - s 4 + s 6   p = - s 4 + s 8


 
60 . Coloring, {8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 7 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y6, y5, y4, 0, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y5, y1, 2 y1 + 4 y5 - 3 y3, 0, 0, -3 y5 + 2 y3, y5, y6, y4, y3, 0, y2]

  p = - s 5 + s 9   p' = - s 5 + s 7   p = - s 5 + s 7


 
61 . Coloring, {8, 12}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, 0, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y2, 2 y4, y3, y4, y5, y6, 0, y8, 0, y7]

  p = - s 5 + s 9


 
62 . Coloring, {9, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y4, y3, y5, 0, 0, y7, y6, 2 y3]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
63 . Coloring, {9, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 0, 0, y7, y5, y6]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y7, 2 y5, y8, y6, 0]

  p = - s 4 + s 9


 
64 . Coloring, {9, 12}

Ωp(Δ)=0:     p = 6s 2 - s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[7 y5, 0, 0, 0, 0, 9 y5 + 9 y6 - 7 y1 - 7 y2 + 9 y3 - 7 y4, 7 y6, 0, 7 y1, 7 y2, 7 y3, 7 y4]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y5, 2 y4, y4, y3, y8, 0, y7, y6, 0]

  p = - s 3 + s 9


 
65 . Coloring, {10, 11}

Ωp(Δ)=0:     p = s 2 + 3s 3 - 6s 4 + 8s 5 - 24s 6 + 32s 7 - 32s 8

R: [7, 7, 7, 6, A, A, B, C, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y2 - y6 + y7 + y3 - y4 + y5, 0, y1, y2, y6, y7, 0, 0, y3, y4, y5]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-y1 - y2 + 2 y4 + 2 y5 - y3 - y6, 0, y1, 0, 0, y2, y4 + y5 - y7, y3, y4, y5, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


M              \ ;       N

$ [ [0, 0, 0, 0, 0, 0, 1164, 0, 1260, 1812, 0, 1472] , [0, 0, 0, 956, 0, 0, 2328, 0, 0, 1539, 0, 885] , [0, 0, 0, 0, 0, 0, 1212, 0, 1284, 1305, 0, 1907] , [0, 956, 0, 0, 1260, 1164, 0, 0, 0, 0, 2328, 0] , [0, 0, 0, 1260, 0, 0, 2132, 0, 0, 1284, 0, 1032] , [0, 0, 0, 1164, 0, 0, 478, 0, 736, 1794, 0, 1536] , [1164, 2328, 1212, 0, 2132, 478, 0, 478, 0, 0, 3624, 0] , [0, 0, 0, 0, 0, 0, 478, 0, 1396, 1770, 0, 2064] , [1260, 0, 1284, 0, 0, 736, 0, 1396, 0, 0, 1032, 0] , [1812, 1539, 1305, 0, 1284, 1794, 0, 1770, 0, 0, 1912, 0] , [0, 0, 0, 2328, 0, 0, 3624, 0, 1032, 1912, 0, 2520] , [1472, 885, 1907, 0, 1032, 1536, 0, 2064, 0, 0, 2520, 0] ] $     $ [ [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0] , [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0] , [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0] , [1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0] , [0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0] ] $

τ= 74 , r'= 1/2

R: [7, 7, 7, 6, A, A, B, C, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, B, C, C, 1, 9]

Ranges

Action of R on ranges, [[24], [24], [6], [16], [19], [24], [6], [16], [24], [24], [6], [16], [21], [21], [14], [29], [7], [17], [29], [29], [7], [17], [30], [15], [30], [8], [18], [15], [5], [13]]
Action of B on ranges, [[21], [22], [22], [20], [23], [26], [27], [25], [26], [27], [27], [25], [9], [9], [1], [11], [12], [10], [11], [12], [12], [10], [29], [3], [30], [30], [28], [4], [4], [2]]

Cycles:    R , {{2, 4, 6, 7, 10, 11}},   B , {{9, 12}, {1, 3, 6, 8, 11}}

β({1, 7}) = 291/11416
β({1, 9}) = 315/11416
β({1, 10}) = 453/11416
β({1, 12}) = 46/1427
β({2, 4}) = 239/11416
β({2, 7}) = 291/5708
β({2, 10}) = 1539/45664
β({2, 12}) = 885/45664
β({3, 7}) = 303/11416
β({3, 9}) = 321/11416
β({3, 10}) = 1305/45664
β({3, 12}) = 1907/45664
β({4, 5}) = 315/11416
β({4, 6}) = 291/11416
β({4, 11}) = 291/5708
β({5, 7}) = 533/11416
β({5, 10}) = 321/11416
β({5, 12}) = 129/5708
β({6, 7}) = 239/22832
β({6, 9}) = 23/1427
β({6, 10}) = 897/22832
β({6, 12}) = 48/1427
β({7, 8}) = 239/22832
β({7, 11}) = 453/5708
β({8, 9}) = 349/11416
β({8, 10}) = 885/22832
β({8, 12}) = 129/2854
β({9, 11}) = 129/5708
β({10, 11}) = 239/5708
β({11, 12}) = 315/5708

Partitions
α([{1, 2, 3, 5, 6, 8, 11}, {4, 7, 9, 10, 12}]) = 1/1

b1 = {1, 2, 3, 5, 6, 8, 11} ` , ` b2 = {4, 7, 9, 10, 12}

Action of R and B on the blocks of the partitions: = [2, 1] [1, 2]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {10, 11}
Rank2
R,B [7, 7, 7, 6, A, A, B, C, B, 2, 4, 5], [6, 8, 8, 7, 3, 3, A, B, C, C, 1, 9]
π2 [0, 0, 0, 0, 0, 1164, 0, 1260, 1812, 0, 1472, 0, 956, 0, 0, 2328, 0, 0, 1539, 0, 885, 0, 0, 0, 1212, 0, 1284, 1305, 0, 1907, 1260, 1164, 0, 0, 0, 0, 2328, 0, 0, 2132, 0, 0, 1284, 0, 1032, 478, 0, 736, 1794, 0, 1536, 478, 0, 0, 3624, 0, 1396, 1770, 0, 2064, 0, 1032, 0, 1912, 0, 2520]
u2 [0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1] (dim 1)
wpp [7, 7, 7, 5, 7, 7, 5, 7, 5, 5, 7, 5]

 

 
66 . Coloring, {10, 12}

R: [7, 7, 7, 6, A, A, B, C, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y6, y4, 0, y5, y5, y3, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y4, y3, y7, y8, 0, y6, y5, y9]


 
67 . Coloring, {11, 12}

R: [7, 7, 7, 6, A, A, B, C, B, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 8 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y7, 0, y3, y4, y5, y6]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y6, y4, y8, y5, 0, y6, y7, y8]

  p = - s 4 + s 9   p' = - s 4 + s 9


 
68 . Coloring, {2, 3, 4}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, B, C, B, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 2 y2, 0, -7 y1 + 3 y2 + 4 y4, -27 y1 + 7 y2 - 2 y3 + 16 y4, 0, 2 y3, -16 y1 + 4 y2 + 10 y4, 2 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 3 y3, 3 y4, 0, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
69 . Coloring, {2, 3, 5}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, C, 1, 5]
B: [6, 7, 7, 7, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[6 y1 - 5 y3 + 6 y4 + 6 y5 - 5 y6, 0, 5 y1 - 5 y2 + 5 y4 + 5 y5 - 5 y7, 0, 5 y1, 5 y2, 5 y3, 5 y4, 0, 5 y5, 5 y6, 5 y7]

  p = - s 3 - s 5 + s 6 + s 8   p' = - s 3 - s 5 + s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -6 y1 - 3 y5 + 13 y2 - 3 y3 - 3 y4 + 13 y6, 3 y1 - 3 y4, 3 y1, 0, 3 y4, 3 y5, 0, 3 y2, 3 y3, 3 y4, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 9   p = s 3 - s 5 - s 6 + s 8


 
70 . Coloring, {2, 3, 6}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, C, 1, 5]
B: [6, 7, 7, 7, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[5 y5 + 6 y2 - 8 y3 - 8 y1 + 5 y6, 0, -y4 + 4 y5 + 5 y2 - 6 y3 - 6 y1 + 4 y6, 0, y4, y5, y2, y3, 0, y1, -2 y5 - 4 y2 + 5 y3 + 5 y1 - 2 y6, y6]

  p = s 4 - s 7   p' = s 4 - s 7   p' = s 5 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y6 + 13 y3 - 3 y4 - 9 y5 + 13 y2, 3 y5, 3 y1, 0, 3 y5, 3 y6, 0, 3 y3, 3 y4, 3 y5, 3 y2]

  p = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 9   p = s 3 - s 5 - s 6 + s 8


 
71 . Coloring, {2, 3, 7}

R: [7, 8, 8, 6, A, A, A, C, B, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y5, y3, 2 y5, 0, y4, y5, y6]

  p = - s 4 + s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 6 y2, -3 y1 - 9 y2 - 3 y3 + 13 y4 - 3 y5 + 13 y6, 3 y1, 0, 3 y2, 3 y3, 0, 3 y4, 0, 3 y5, 3 y6]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 + s 4 - s 6 - s 7


 
72 . Coloring, {2, 3, 8}

Ωp(Δ)=0:     p = 2s 2 - s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 4 vs 8 5 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y4, 0, 0, 0, y3, -10 y4 - 7 y3 + 3 y2 + 16 y1, y2, -20 y4 - 14 y3 + 6 y2 + 32 y1, 0, y1, -8 y4 - 4 y3 + 2 y2 + 13 y1, -11 y4 - 8 y3 + 4 y2 + 18 y1]

  p = - s 2 + s 5   p = - s 2 + s 8   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -y1 - 3 y3 - y2 + 6 y5 - y4, y1, 2 y3, 0, y3, y2, 0, y5, y4, 0, y5]

  p' = s 4 - s 7   p = - s 3 + s 6   p' = s 3 - s 6

 ยป SYNC'D !RANK'D


 
73 . Coloring, {2, 3, 9}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 8 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[-4 y1 - 2 y2 + 2 y3 + 2 y4, 0, 0, 0, 16 y1 + 9 y2 - 11 y3 - 2 y4, 2 y1, 14 y1 + 7 y2 - 9 y3 - 2 y4, 4 y1, 0, 2 y2, 2 y3, 2 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p = - s 2 + s 8   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y6, y4, 0, 2 y6, y5, 2 y2 - 2 y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
74 . Coloring, {2, 3, 10}

R: [7, 8, 8, 6, A, A, B, C, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 5 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[3 y7, 3 y8, 0, 0, 3 y5, 3 y6, 3 y4, 3 y3, 0, 3 y2, 3 y1, 5 y7 - 3 y8 - 3 y5 - 3 y6 + 5 y4 - 3 y3 - 3 y2 + 5 y1]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, y1, 0, y4, 2 y1 - 2 y4 + y2 + y3 - y5, 0, y2, y3, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8


 
75 . Coloring, {2, 3, 11}

R: [7, 8, 8, 6, A, A, B, C, B, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-3 y2 - 3 y1 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y2, 3 y1, 0, 0, 3 y3, 3 y4, 0, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
76 . Coloring, {2, 3, 12}

R: [7, 8, 8, 6, A, A, B, C, B, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, 2 y2, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y5, y4, y4, y6, y3, 0, 0, y2, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
77 . Coloring, {2, 4, 5}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, C, 1, 5]
B: [6, 7, 8, 6, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y7, 0, y7, y5, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, -3 y3 + 6 y7 - 3 y6 + 6 y8, -3 y1 - 3 y4 - 3 y2 + 7 y7 - 3 y5 + 7 y8, 3 y1, 0, 3 y4, 3 y3, 3 y2, 3 y7, 3 y6, 3 y5, 3 y8]

  p = - 2s - 3s 2 - 2s 3 + s 5 + 2s 6 + 3s 7 + 2s 8 - s 10   p = s + 2s 2 + 2s 3 + s 4 - s 6 - 2s 7 - 2s 8 - s 9


 
78 . Coloring, {2, 4, 6}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, C, 1, 5]
B: [6, 7, 8, 6, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[3 y3, 0, 3 y4, 0, 7 y3 + 4 y4 - 3 y1 - 3 y2, 0, 3 y3 + 3 y4, 3 y4, 0, 3 y1, 3 y3 + 3 y4, 3 y2]

  p' = s 4 - s 7   p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y2, 3 y1, -3 y2 - 3 y1 - 3 y9 - 3 y8 - 3 y7 + 13 y6 - 3 y5 - 3 y4 + 13 y3, 0, 3 y9, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3]

  p = - s 6 - s 7 + s 9 + s 10


 
79 . Coloring, {2, 4, 7}

R: [7, 8, 7, 7, A, A, A, C, B, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y5, 0, y6, y3, 0, y2, y3, y4]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
80 . Coloring, {2, 4, 8}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-y4 - y1 - y2 + 5 y3, 0, 0, 0, y3, 0, y4, y1, 0, y3, y2, y3]

  p = s 2 - s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 5 y8, 5 y7, 5 y6, 0, 5 y5, 5 y4, 5 y3, 5 y2, 5 y1, 0, 11 y8 - 5 y7 - 5 y6 - 5 y5 + 11 y4 - 5 y3 - 5 y2 + 11 y1]

  p = s 5 + s 6 - s 8 - s 9


 
81 . Coloring, {2, 4, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-7 y1 + 3 y2 + 3 y3 + 4 y4, 0, 0, 0, -27 y1 + 7 y2 + 7 y3 + 16 y4, 0, 2 y1, 2 y2, 0, 2 y3, -16 y1 + 4 y2 + 4 y3 + 10 y4, 2 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, 5 y1, 11 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y7 - 5 y8, 5 y2, 0, 5 y3, 5 y4, 5 y5, 5 y6, 5 y7, 5 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
82 . Coloring, {2, 4, 10}

R: [7, 8, 7, 7, A, A, B, C, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[y1 + y2 - y3 + y4 + y5 - y6 + y7, y1, 0, 0, y2, 0, y3, y4, 0, y5, y6, y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, y2, y3, 0, y1, y6 + y7 - y4, -y2 - y3 - y1 + 2 y6 + 2 y7 - y5, y6, y7, y5, y4]

  p' = s 3 - s 8   p = s 3 - s 8


 
83 . Coloring, {2, 4, 11}

R: [7, 8, 7, 7, A, A, B, C, B, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y1, y1, 0, y2, y3, 0, y2 - y3, y4, y4]

  p' = - s 3 + s 6   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[3 y3, 3 y2, 3 y1, 0, 0, -3 y3 - 3 y1 - 3 y4 + 8 y5 - 3 y6 + 8 y8, -3 y2 + 5 y5 - 3 y7 + 5 y8, 3 y4, 3 y5, 3 y7, 3 y6, 3 y8]

  p' = - s - 2s 2 - 2s 3 - s 4 + s 6 + 2s 7 + 2s 8 + s 9   p = - s - 2s 2 - 2s 3 - s 4 + s 6 + 2s 7 + 2s 8 + s 9


 
84 . Coloring, {2, 4, 12}

R: [7, 8, 7, 7, A, A, B, C, B, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y6, y5, y4, 2 y5, y3, y2]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 5 y1, 11 y1 - 5 y2 - 5 y3 - 5 y4 + 11 y6 - 5 y5 + 11 y7 - 5 y8 - 5 y9, 5 y2, 5 y3, 5 y4, 5 y6, 5 y5, 0, 5 y7, 5 y8, 5 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10


 
85 . Coloring, {2, 5, 6}

R: [7, 8, 7, 6, 3, 3, B, C, B, C, 1, 5]
B: [6, 7, 8, 7, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y7, y6, y5, y6, 0, 0, y4, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 6 y3 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 0, 3 y1, 0, 3 y3, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = s 4 + s 5 - s 7 - s 8   p = s 4 - s 6 - s 7 + s 9


 
86 . Coloring, {2, 5, 7}

Ωp(Δ)=0:     p = 3s 4 + 4s 5 + 8s 7 + 16s 8

R: [7, 8, 7, 6, 3, A, A, C, B, C, 1, 5]
B: [6, 7, 8, 7, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y4, 0, y2, y4, y1]

  p = - s 3 + s 8   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1 + 3 y4, 3 y1, -6 y1 - 9 y4 - 3 y2 - 3 y3 + 13 y7 - 3 y5 + 13 y6, 0, 3 y4, 3 y2, 3 y3, 3 y7, 3 y4, 3 y5, 3 y6]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9   p = - s 4 + s 10


 
87 . Coloring, {2, 5, 8}

R: [7, 8, 7, 6, 3, A, B, B, B, C, 1, 5]
B: [6, 7, 8, 7, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y3, 0, y2, 0, y1, y7, y8, y7, 0, y6, y5, y4]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, 5 y1, 5 y3, -14 y1 + 18 y3 - 14 y2 + 18 y4 + 18 y5 - 14 y6 + 18 y7, 0, -7 y1 + 9 y3 - 7 y2 + 9 y4 + 9 y5 - 7 y6 + 9 y7, 5 y2, 5 y4, 5 y5, 5 y6, 0, 5 y7]

  p = s 4 - s 6 - s 7 + s 9   p = - s 4 - s 5 + s 7 + s 8


 
88 . Coloring, {2, 5, 9}

R: [7, 8, 7, 6, 3, A, B, C, C, C, 1, 5]
B: [6, 7, 8, 7, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y7, y4, y3, y4, 0, y5, y6, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, y2, y1, 0, y6, y4, y5, 2 y6, y7, y8, 0]

  p = - s 6 + s 9


 
89 . Coloring, {2, 5, 10}

R: [7, 8, 7, 6, 3, A, B, C, B, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y3, y2, y1, 0, y9, y8, y6, y7, 0, y4, y5, y10]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y7, y4, y8, y5, y6, y9]


 
90 . Coloring, {2, 5, 11}

R: [7, 8, 7, 6, 3, A, B, C, B, C, 4, 5]
B: [6, 7, 8, 7, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 9 8 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1 - y2 - y3 + y4, y1 - y2 - y3 + y4, y1, y2 + y3, y2, 0, y3, y4, y4]

  p = - s 2 + s 3 - s 4 + s 9   p = - s 2 + s 3 - s 4 + s 5   p = - s 2 + s 6   p' = - s 2 + s 6   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}, {9, 12}}   

See Matrix
 

[-3 y1 - 3 y2 - 3 y4 + 6 y8 - 3 y6 + 6 y7, -3 y3 + 7 y8 - 3 y5 + 7 y7, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y8, 3 y5, 3 y6, 3 y7]

  p = - s - 2s 2 - 2s 3 - s 4 + s 6 + 2s 7 + 2s 8 + s 9   p = 2s + 3s 2 + 2s 3 - s 5 - 2s 6 - 3s 7 - 2s 8 + s 10


 
91 . Coloring, {2, 5, 12}

R: [7, 8, 7, 6, 3, A, B, C, B, C, 1, 9]
B: [6, 7, 8, 7, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y3, y2, y3, y7, y4, y5, y6]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y4, y3, y4 + y7, y7, y2, y6, 0, y5, y8, y7]

  p' = - s 6 + s 9   p = - s 6 + s 9


 
92 . Coloring, {2, 6, 7}

R: [7, 8, 7, 6, A, 3, A, C, B, C, 1, 5]
B: [6, 7, 8, 7, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y3 + y6, 0, y3, 0, y4, y6, y5, y6, 0, y2, y6, y1]

  p = s 4 - s 7   p' = s 5 - s 8   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 6 y2 - 3 y3 - 6 y7 + 13 y6 - 3 y5 + 13 y4, 3 y2, 3 y1, 0, 3 y2, 3 y3, 3 y7, 3 y6, 3 y7, 3 y5, 3 y4]

  p' = s 5 + s 6 - s 8 - s 9   p = s 4 - s 10   p' = s 4 - s 6 - s 7 + s 9


 
93 . Coloring, {2, 6, 8}

R: [7, 8, 7, 6, A, 3, B, B, B, C, 1, 5]
B: [6, 7, 8, 7, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[3 y1, 0, 3 y2, 0, 3 y3, -7 y1 - 7 y2 + 14 y3 + 8 y5 - 3 y4, -6 y3 + 6 y5 + 3 y4, -7 y1 - 7 y2 + 14 y3 + 8 y5 - 3 y4, 0, 3 y5, 3 y4, -5 y1 - 5 y2 + 7 y3 + 7 y5]

  p' = s 3 - s 6   p = - s 3 + s 6   p' = s 4 - s 7   p = - s 3 + s 9

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 3 y1 + 3 y3 - 10 y4 + 3 y5, 6 y1 + 6 y3 - 20 y4 + 6 y5, 0, 3 y1 + 3 y3 - 10 y4 + 3 y5, y3, 3 y1 + 3 y3 - 9 y4 + 3 y5 - y2, y2, y5, 0, y4]

  p = - s 3 + s 9   p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6


 
94 . Coloring, {2, 6, 9}

R: [7, 8, 7, 6, A, 3, B, C, C, C, 1, 5]
B: [6, 7, 8, 7, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-y1 - y3 + y5 + y2, 0, y1, 0, -2 y3 + 2 y5 + 2 y2 - y4, y3 - 2 y5 + y4, y2, y3 - 2 y5 + y4, 0, y3, y5, y4]

  p = - s 3 + s 6   p = - s 3 + s 9   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y3, y4, y6, 2 y3, y7, y5, 0]

  p = s 5 - s 8   p' = s 5 - s 8


 
95 . Coloring, {2, 6, 10}

R: [7, 8, 7, 6, A, 3, B, C, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[7 y4, 7 y1, -7 y4 + 9 y1 + 9 y2 - 7 y3 - 7 y5 + 9 y6 + 9 y9 - 7 y7 + 9 y8, 0, 7 y2, 7 y3, 7 y5, 7 y6, 0, 7 y9, 7 y7, 7 y8]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, 2 y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, y2, y3, y4, y5, y6, y7]

  p = - s 7 + s 8   p = - s 7 + s 9


 
96 . Coloring, {2, 6, 11}

R: [7, 8, 7, 6, A, 3, B, C, B, C, 4, 5]
B: [6, 7, 8, 7, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -7 y1 + 9 y2 - 7 y3 - 7 y4 + 9 y5 + 9 y6 - 7 y7 + 9 y8, 7 y1, 7 y2, 7 y3, 7 y4, 7 y5, 0, 7 y6, 7 y7, 7 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y8 - 3 y6 - 3 y7 + 13 y9, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5, 3 y8, 3 y6, 3 y7, 3 y9]

  p = - s 6 - s 7 + s 9 + s 10


 
97 . Coloring, {2, 6, 12}

R: [7, 8, 7, 6, A, 3, B, C, B, C, 1, 9]
B: [6, 7, 8, 7, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, y4, y3, y4, y2, y4, y1, 2 y5 + y4]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y5, y3, y4, y2, y9, y6, y1, 0, y7, y8, y9]

  p = - s 7 + s 10


 
98 . Coloring, {2, 7, 8}

R: [7, 8, 7, 6, A, A, A, B, B, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y4, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y1 - 2 y4 - 2 y5 + 2 y3, y1, 2 y4 + 2 y5 - y6 - y2, 0, y1 - y4 - y5 + y3, y6, y4, y5, 0, y2, y3]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
99 . Coloring, {2, 7, 9}

R: [7, 8, 7, 6, A, A, A, C, C, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, y3, y2, y3, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y4, y1, y2, 0, y4, y3, y5, 2 y4, 0, y6, 0]

  p = s 4 - s 7   p' = s 4 - s 7


 
100 . Coloring, {2, 7, 10}

R: [7, 8, 7, 6, A, A, A, C, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, y7, y4, y5, 0, y6, y7, y8]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 5 y2, 5 y3, 0, -5 y2 - 5 y3 - 5 y1 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 5 y1, 5 y4, 5 y5, 0, 5 y6, 5 y7]

  p = - s 4 - s 5 + s 7 + s 8


 
101 . Coloring, {2, 7, 11}

R: [7, 8, 7, 6, A, A, A, C, B, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, y2, y1, 2 y6, y6, 0, y5, y6, y4]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8]

  p = s 3 + s 4 - s 8 - s 9


 
102 . Coloring, {2, 7, 12}

R: [7, 8, 7, 6, A, A, A, C, B, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, y6, y7, y5, y4, y3, y2]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y7, y2, y1, y5, y7, y3, y4, 0, 0, y6, y7]

  p' = s 5 - s 8   p = - s 5 + s 8


 
103 . Coloring, {2, 8, 9}

R: [7, 8, 7, 6, A, A, B, B, C, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y2, 0, 0, 0, -y1 + y3, y1, -y1 + y4, y1, 0, y2, y3, y4]

  p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8   p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y5, 0, y3, y6, y4, y7, y8, y9, y10]


 
104 . Coloring, {2, 8, 10}

R: [7, 8, 7, 6, A, A, B, B, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 2 y3, y3, y4, y6, 0, y7, y5, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y2, 0, y2, 2 y1 - 2 y2, y3, y4, -2 y1 + 4 y2 + 2 y3, 0, -5 y1 + 9 y2 + 3 y3 + y4]

  p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8   p = - s 4 + s 5


 
105 . Coloring, {2, 8, 11}

R: [7, 8, 7, 6, A, A, B, B, B, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y5, y6, 2 y7, y7, 0, y2, y3, y4]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, y4, 0, 0, y3, y5, y2, y1, y5, 0, -y6 + 5 y5 - y4 - y3 - y2 - y1]

  p' = s 6 - s 8   p' = s 5 - s 7   p = s 5 - s 9


 
106 . Coloring, {2, 8, 12}

R: [7, 8, 7, 6, A, A, B, B, B, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y2, y1, y2, y3, y4, y6, y5]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 2 y1 - y4 + 2 y3 - y5, y1 + y2 + y3 - y6, 2 y2, y1, y2, y4, y3, 0, y5, 0, y6]

  p = - s 2 + s 5 + s 6 - s 9   p = - s 2 - s 3 + 2s 6 + s 8 - s 9   p' = - s 3 - s 5 + s 6 + s 8


 
107 . Coloring, {2, 9, 10}

R: [7, 8, 7, 6, A, A, B, C, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[y4, y6, 0, 0, 5 y4 - y6 - y1 - y2 - y3 - y5, y1, y4, y2, 0, y3, y4, y5]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1 + y2 - y3 + y4 + y5 + y6 - y7 - y8, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
108 . Coloring, {2, 9, 11}

R: [7, 8, 7, 6, A, A, B, C, C, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, y5, y7, 2 y2, y2, 0, y3, y1, y4]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[5 y5 - y1 - y2 - y3 - y4 - y6, y5, y1, 0, 0, y2, y5, y3, y4, y5, y6, 0]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8


 
109 . Coloring, {2, 9, 12}

R: [7, 8, 7, 6, A, A, B, C, C, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 3 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y2, 0, 0, 0, 0, 5 y2 - 2 y3, 2 y2, 5 y2 - 2 y3, -2 y1 + 2 y3, 2 y1, 2 y2, 2 y3]

  p = - s 3 + s 6   p = - s 3 + s 7   p = - s 3 + s 8   p = - s 3 + s 5   p = - s 3 + s 4

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, 2 y4, y4, y5, y6, 0, y8, y7, 0]

  p = - s 6 + s 9


 
110 . Coloring, {2, 10, 11}

R: [7, 8, 7, 6, A, A, B, C, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[7 y1, 0, 7 y2, 0, 0, 7 y3, 9 y1 + 9 y2 + 9 y3 + 9 y4 - 7 y7 - 7 y6 + 9 y5 - 7 y8, 7 y4, 7 y7, 7 y6, 7 y5, 7 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
111 . Coloring, {2, 10, 12}

R: [7, 8, 7, 6, A, A, B, C, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, y3, 0, y2, y1, -y8 + y7 + y6 + y5 - y4 + y3 + y2 - y1]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
112 . Coloring, {2, 11, 12}

R: [7, 8, 7, 6, A, A, B, C, B, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, 0, y5, 2 y3, y3, y4, y2, y1, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[5 y5 - y2 - y1 - y3 - y4 - y6 - y7, y5, y2, 0, y1, y3, y5, y4, 0, y5, y6, y7]

  p' = - s 3 + s 8   p' = - s 4 + s 9   p = - s 3 + s 8


 
113 . Coloring, {3, 4, 5}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, C, 1, 5]
B: [6, 8, 7, 6, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[2 y2 + 2 y3 + 2 y6 - y1 - y5, 0, y2 + y3 + y6 - y4, 0, y2, 0, y1, y3, 0, y6, y5, y4]

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, -3 y2 - 3 y3 + 5 y6 - 3 y7 + 8 y8, -3 y1 - 3 y5 + 8 y6 - 3 y4 + 5 y8, 3 y1, 0, 3 y2, 3 y3, 3 y5, 3 y6, 3 y4, 3 y7, 3 y8]

  p = - s + s 9   p' = - s + s 9


 
114 . Coloring, {3, 4, 6}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, C, 1, 5]
B: [6, 8, 7, 6, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y2, 0, y1, 0, -y1 + y4, 0, y3, y2 - y5, 0, y5, y4, y3]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y8, 3 y7, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y7 - 3 y6 - 3 y3 + 8 y2 + 5 y1, -3 y8 - 3 y5 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = s 3 - s 9   p = s 3 - s 9


 
115 . Coloring, {3, 4, 7}

R: [7, 7, 8, 7, A, A, A, C, B, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, y4, 0, y6, y4, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
116 . Coloring, {3, 4, 8}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[5 y4 - y1 - y2 - y3, 0, 0, 0, y4, 0, y1, y2, 0, y4, y3, y4 ]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y5, y1, y2, 0, y6, y3, y4, y7, y8, 0, y9]


 
117 . Coloring, {3, 4, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[10 y2 - 16 y4 + 4 y3, 0, 0, 0, 2 y2, 0, 4 y2 - 7 y4 + 3 y3, 2 y1, 0, 16 y2 - 2 y1 - 27 y4 + 7 y3, 2 y4, 2 y3]

  p' = - s 3 + s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y8, y6, y7, 0, y4, y5, y3, y1, y2, y9, 0]


 
118 . Coloring, {3, 4, 10}

R: [7, 7, 8, 7, A, A, B, C, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, y1, 0, 0, y2, 0, y3, y4, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
119 . Coloring, {3, 4, 11}

R: [7, 7, 8, 7, A, A, B, C, B, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y1, y1, 0, y2 + y3, y2, 0, y3, y4, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[-3 y1 - 3 y4 + 8 y5 - 3 y6 + 5 y8, -3 y2 - 3 y3 + 5 y5 - 3 y7 + 8 y8, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s + s 9   p' = - s + s 9


 
120 . Coloring, {3, 4, 12}

R: [7, 7, 8, 7, A, A, B, C, B, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, 2 y3, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y1 + y2 - y3 - y4 - y5 + y7 + y6 - y9 + y8, y1, y2, y3, y4, y5, y7, 0, y6, y9, y8]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10


 
121 . Coloring, {3, 5, 6}

R: [7, 7, 8, 6, 3, 3, B, C, B, C, 1, 5]
B: [6, 8, 7, 7, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[7 y1, 0, 7 y2, 0, 9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 + 9 y6 - 7 y7, 7 y3, 7 y4, 7 y5, 0, 0, 7 y6, 7 y7]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y4 + 5 y2 - 3 y6 + 8 y5, 0, -3 y3 + 8 y2 - 3 y7 + 5 y5, 0, 3 y1, 3 y4, 3 y3, 3 y2, 3 y7, 3 y6, 3 y5]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
122 . Coloring, {3, 5, 7}

R: [7, 7, 8, 6, 3, A, A, C, B, C, 1, 5]
B: [6, 8, 7, 7, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 7 vs 10

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y4 - y3 + y5, 0, -y2 + y4, 0, y3, y5, y2, y1, 0, y4 + y5 - y1, y5, y4]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8   p = - s 5 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y1, 3 y1 - 3 y5, -6 y1 - 3 y5 - 3 y2 - 3 y3 + 13 y4 - 3 y6 + 13 y7, 0, 3 y5, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 + s 6 + s 7 - s 9   p = - s 4 - s 5 + s 7 + s 8   p = - s 4 + s 10


 
123 . Coloring, {3, 5, 8}

R: [7, 7, 8, 6, 3, A, B, B, B, C, 1, 5]
B: [6, 8, 7, 7, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, 0, y4, 0, y3, y2, y7, y8, 0, y5, y6, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1, y2, 2 y3, 0, y3, y4, y5, y8, y6, 0, y7]

  p = - s 7 + s 9


 
124 . Coloring, {3, 5, 9}

R: [7, 7, 8, 6, 3, A, B, C, C, C, 1, 5]
B: [6, 8, 7, 7, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[y4, 0, -y1 - y2 - y3 - y5 + 5 y4 - y6, 0, y1, y2, y4, y3, 0, y5, y4, y6]

  p' = - s 3 + s 7   p = - s 3 + s 7   p' = - s 4 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y8, y7, y6, 0, y5, y4, y3, 2 y5, y2, y1, 0]

  p = s 3 - s 9


 
125 . Coloring, {3, 5, 10}

R: [7, 7, 8, 6, 3, A, B, C, B, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[-5 y1 + 11 y2 + 11 y3 - 5 y4 - 5 y5 + 11 y6 - 5 y7 - 5 y8 + 11 y9, 5 y1, 5 y2, 0, 5 y3, 5 y4, 5 y5, 5 y6, 0, 5 y7, 5 y8, 5 y9]

  p = s 4 + s 5 + s 6 - s 8 - s 9 - s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y7, y1, 0, y5, y4, y5, y3, y2, y7, y6]

  p' = s 6 - s 8   p = s 6 - s 8


 
126 . Coloring, {3, 5, 11}

R: [7, 7, 8, 6, 3, A, B, C, B, C, 4, 5]
B: [6, 8, 7, 7, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y8, y7, y4, y5, y6, y8 - y7 - y4 + y5 - y6 - y1 + y2 + y3, 0, y1, y2, y3]

  p = - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}, {9, 12}}    order:   8

See Matrix
 

[-3 y1 - 3 y3 + 8 y7 - 3 y6 + 5 y8, -3 y4 - 3 y2 + 5 y7 - 3 y5 + 8 y8, 3 y1, 0, 0, 3 y4, 3 y2, 3 y3, 3 y7, 3 y6, 3 y5, 3 y8]

  p = - s + s 9   p' = - s + s 9


 
127 . Coloring, {3, 5, 12}

R: [7, 7, 8, 6, 3, A, B, C, B, C, 1, 9]
B: [6, 8, 7, 7, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y3, y2, y5, y4, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y7 - y3 + y4 + y5 - y6, y2 - y7, y1, y2, y7, y3, y4, 0, y5, y6, y7]

  p = s 3 - s 9   p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p' = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
128 . Coloring, {3, 6, 7}

R: [7, 7, 8, 6, A, 3, A, C, B, C, 1, 5]
B: [6, 8, 7, 7, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2 - y3 + y5, 0, y2 - y3, 0, y1, y5, y2, y3, 0, y4, y5, y6]

  p' = - s 5 + s 8   p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y7, 3 y3, -3 y7 - 6 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y2 - 3 y1 + 13 y8, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y2, 3 y1, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9   p = s 5 - s 7 - s 8 + s 10


 
129 . Coloring, {3, 6, 8}

R: [7, 7, 8, 6, A, 3, B, B, B, C, 1, 5]
B: [6, 8, 7, 7, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-3 y1 + 14 y2 - 7 y3 - 7 y5 + 8 y6, 0, 3 y1, 0, 3 y2, 3 y3, 8 y2 - 7 y3 - 3 y4 - 7 y5 + 14 y6, 3 y4, 0, 7 y2 - 5 y3 - 5 y5 + 7 y6, 3 y5, 3 y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y2, y3, 2 y3, 0, y3, y1, y4, y5, y6, 0, y7]

  p' = - s 6 + s 8   p = - s 6 + s 8


 
130 . Coloring, {3, 6, 9}

R: [7, 7, 8, 6, A, 3, B, C, C, C, 1, 5]
B: [6, 8, 7, 7, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y4, 0, -y1 - y2 - y5 - y3 + 5 y4 - y6, 0, y1, y2, y4, y5, 0, y3, y4, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y3, y4, y5, 2 y3, y6, y7, 0]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
131 . Coloring, {3, 6, 10}

R: [7, 7, 8, 6, A, 3, B, C, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 6 + s 8   p' = - s 6 + s 8


 
132 . Coloring, {3, 6, 11}

R: [7, 7, 8, 6, A, 3, B, C, B, C, 4, 5]
B: [6, 8, 7, 7, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y9, y7, y8, y6, y5, y4, 0, y3, y1, y2]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[3 y1, -3 y4 - 3 y5 + 5 y2 - 3 y6 + 8 y7, -3 y1 - 3 y3 + 8 y2 - 3 y8 + 5 y7, 0, 0, 3 y4, 3 y5, 3 y3, 3 y2, 3 y8, 3 y6, 3 y7]

  p' = s 3 - s 9   p = - s 3 + s 9


 
133 . Coloring, {3, 6, 12}

R: [7, 7, 8, 6, A, 3, B, C, B, C, 1, 9]
B: [6, 8, 7, 7, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y4, y5, y3, y6, y4, y8, y7]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y2 - y3 - y5 + y4 + y6 - y7, y1, y2, y3, y8, y5, y4, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9   p = - s 4 + s 10


 
134 . Coloring, {3, 7, 8}

R: [7, 7, 8, 6, A, A, A, B, B, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y6, y7, y2, y7, 0, y3, y4, y5]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 14 y2, 7 y1, 11 y1 - 16 y2 - 7 y3 + 9 y4 - 7 y5 + 9 y6, 0, 7 y2, 7 y3, 14 y1 - 21 y2, 7 y4, 0, 7 y5, 7 y6]

  p = s 3 - s 5 - s 6 + s 8   p = - s 3 + s 9   p = - s 3 - s 4 + s 6 + s 7


 
135 . Coloring, {3, 7, 9}

R: [7, 7, 8, 6, A, A, A, C, C, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, y3, y2, y3, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y5, y2, y1, 0, y5, y4, 2 y2 - 3 y5, 2 y5, 0, y3, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7


 
136 . Coloring, {3, 7, 10}

R: [7, 7, 8, 6, A, A, A, C, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, 0, 0, y4, y3, y2, y3, 0, y1, y3, y6 - y3]

  p = - s 4 + s 7   p' = s 5 - s 8   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -5 y1 - 10 y3 - 5 y2 + 11 y4 - 5 y5 + 11 y6, 5 y1, 0, 5 y3, 5 y2, 5 y3, 5 y4, 0, 5 y5, 5 y6]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 - s 5 - s 6 + s 8


 
137 . Coloring, {3, 7, 11}

R: [7, 7, 8, 6, A, A, A, C, B, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, y5, y4, 2 y3, y3, 0, y2, y3, y1]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8]

  p = s 3 + s 4 - s 8 - s 9


 
138 . Coloring, {3, 7, 12}

R: [7, 7, 8, 6, A, A, A, C, B, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y3, 0, 0, 0, 0, y2, y1, y2, y4, y6, y5, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y6, y2, y1, y3, y6, y4, 2 y3 - 3 y6, 0, 0, y5, y6]

  p' = - s 5 + s 8   p = - s 4 + s 7   p' = - s 4 + s 7


 
139 . Coloring, {3, 8, 9}

R: [7, 7, 8, 6, A, A, B, B, C, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y2, 0, 0, 0, -y3 + y1, y3, y4, y3, 0, y2, y1, y3 + y4]

  p' = - s 3 + s 6   p = - s 2 + s 8   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, y7, y8, y9, y10]


 
140 . Coloring, {3, 8, 10}

R: [7, 7, 8, 6, A, A, B, B, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y6, y6, y3, y6, 0, y4, y5, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, 0, y4, y3, y4, y2, y6, 0, y5]

  p = - s 5 + s 7   p' = - s 5 + s 7


 
141 . Coloring, {3, 8, 11}

R: [7, 7, 8, 6, A, A, B, B, B, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y6, y5, 2 y4, y4, 0, y3, y2, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, y4, y2, 0, 0, y3, y5, y6, y7, y8, 0, y9]


 
142 . Coloring, {3, 8, 12}

R: [7, 7, 8, 6, A, A, B, B, B, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y1, y2, 2 y3, y4, y3, y5, y6, 0, y7, 0, y8]

  p = - s 2 + s 9


 
143 . Coloring, {3, 9, 10}

R: [7, 7, 8, 6, A, A, B, C, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y4, y5, y6, y5, 0, y3, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y4, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
144 . Coloring, {3, 9, 11}

R: [7, 7, 8, 6, A, A, B, C, C, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, y6, y7, 2 y2, y2, 0, y3, y4, y1]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y3, y2, y1, 0, 0, y6, y5, y4, y9, y8, y7, 0]


 
145 . Coloring, {3, 9, 12}

R: [7, 7, 8, 6, A, A, B, C, C, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 3 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y2, 0, 0, 0, 0, 5 y2 - 2 y3, 2 y2, 5 y2 - 2 y3, -2 y1 + 2 y3, 2 y1, 2 y2, 2 y3]

  p = - s 3 + s 4   p = - s 3 + s 5   p = - s 3 + s 6   p = - s 3 + s 7   p = - s 3 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = - s 3 + s 9


 
146 . Coloring, {3, 10, 11}

R: [7, 7, 8, 6, A, A, B, C, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y4, 0, y1, y2, y3, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
147 . Coloring, {3, 10, 12}

R: [7, 7, 8, 6, A, A, B, C, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y4 + y3 - 2 y6, 0, 0, 0, y6, y5, y6, y4, y3, y2, -y6 + y3]

  p = s 4 - s 7   p' = - s 4 + s 7   p' = s 5 - s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y5, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9


 
148 . Coloring, {3, 11, 12}

R: [7, 7, 8, 6, A, A, B, C, B, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y1, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1 - y2 + y5 + y6 + y7 - y4 - y3 + y8 - y9, y1, y2, 0, y5, y6, y7, y4, 0, y3, y8, y9]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
149 . Coloring, {4, 5, 6}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, C, 1, 5]
B: [6, 8, 8, 6, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, y4, 0, y3, 0, 0, 0, y2, y1]

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, 2 y1, 0, -2 y3 - 2 y4 - 39 y6 + 11 y1 + 11 y2 + 11 y5, 0, 2 y2, 0, 2 y3, 3 y1 + 3 y2 + 3 y5 - 11 y6, 2 y4, 2 y5, 2 y6 ]

  p = - s + s 7   p' = - s + s 7


 
150 . Coloring, {4, 5, 7}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, A, A, C, B, C, 1, 5]
B: [6, 8, 8, 6, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y6, 0, 0, y4, y5, y7]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y8 - 3 y4 - 3 y5 + 13 y6 - 3 y7 - 3 y2 + 13 y3, 3 y1, 3 y8, 0, 3 y4, 0, 3 y5, 3 y6, 3 y7, 3 y2, 3 y3]

  p = - s 3 - s 4 + s 8 + s 9


 
151 . Coloring, {4, 5, 8}

Ωp(Δ)=0:     p' = s 3 - 8s 6 - 16s 7   p = s 3 - 8s 6 - 16s 7

R: [7, 7, 7, 7, 3, A, B, B, B, C, 1, 5]
B: [6, 8, 8, 6, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 2 y6, 2 y5, -4 y6 + 6 y4, 0, 2 y4, 0, 2 y3, 2 y2, -6 y6 + 9 y4, 0, 2 y1]

  p' = s 5 - s 7   p = s 5 - s 7


 
152 . Coloring, {4, 5, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, C, 1, 5]
B: [6, 8, 8, 6, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y4, 0, y3, 0, 0, y7, y6, y5]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y7, 2 y6, 2 y4, 0, 2 y5, 0, 2 y3, 2 y1, 3 y1, 2 y2, 0]

  p = - s 3 + s 8


 
153 . Coloring, {4, 5, 10}

Ωp(Δ)=0:     p = 3s 2 - 2s 3 + 8s 5 - 32s 7   p' = - 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, C, B, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y3, y4, 0, y3, 0, y1, 0, 0, y6, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -y1 - y2 - y3 + y7 + y4 - y5 + y6, y1, 0, y2, 0, y3, y7, y4, y5, y6]

  p = - s 2 - s 3 + s 7 + s 8


 
154 . Coloring, {4, 5, 11}

Ωp(Δ)=0:     p = 3s 2 + 2s 3 - 8s 5 + 32s 7   p' = 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, C, B, C, 4, 5]
B: [6, 8, 8, 6, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y1, y4, 0, y3, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 0, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
155 . Coloring, {4, 5, 12}

Ωp(Δ)=0:     p = 3s 2 - 10s 3 - 40s 5 + 32s 6 - 32s 7 + 128s 8   p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, B, C, B, C, 1, 9]
B: [6, 8, 8, 6, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, 0, y2, 0, y3, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y3, y2, y1, y9, y8, 0, y7, 0, y6, y5, y4]


 
156 . Coloring, {4, 6, 7}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, A, C, B, C, 1, 5]
B: [6, 8, 8, 6, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y3, 0, y2, 0, 0, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, 2 y1, 2 y2, 11 y1 + 11 y2 + 11 y3 - 2 y4 - 39 y6 - 2 y5 + 11 y7, 0, 2 y3, 0, 2 y4, 2 y6, 2 y5, 2 y7, 3 y1 + 3 y2 + 3 y3 - 11 y6 + 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
157 . Coloring, {4, 6, 8}

Ωp(Δ)=0:     p' = s 3 + 4s 4 - 8s 6 - 16s 7   p = s 3 - 16s 5 - 8s 6 + 16s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, B, B, B, C, 1, 5]
B: [6, 8, 8, 6, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, -2 y1 - 2 y4 + 2 y3 + 2 y2, 0, 2 y4, 0, -9 y4 - 2 y3 + 7 y2, 0, 0, 2 y3, 2 y2, -7 y4 - 2 y3 + 5 y2]

  p' = - s 2 + s 5   p = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y2, 2 y2, 0, y3, 0, y4, y5, y6, 0, y7]

  p = - s 6 + s 8


 
158 . Coloring, {4, 6, 9}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, C, 1, 5]
B: [6, 8, 8, 6, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[16 y1, 0, 16 y3, 0, 16 y2, 0, 20 y1 + 20 y3 + 27 y2 - 25 y4, 0, 0, -20 y2 + 28 y4, 16 y4, 28 y1 + 28 y3 + 25 y2 - 35 y4]

  p' = - s 3 + s 6   p = s 2 - s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y7, y6, y5, 0, y4, 0, y3, 2 y6, y1, y2, 0]

  p = - s 2 + s 8


 
159 . Coloring, {4, 6, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, B, C, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y7, 0, y3, 0, y4, 0, 0, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, 0, y6, y4, y5, y7, y8]


 
160 . Coloring, {4, 6, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, B, C, B, C, 4, 5]
B: [6, 8, 8, 6, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -y2 - 6 y1 + 4 y3 + 5 y4, y2, 5 y1 - 2 y3 - 4 y4, 0, y1, 0, 0, -8 y1 + 5 y3 + 6 y4, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y7 + 8 y3 - 3 y4 + 5 y2, -3 y1 - 3 y6 + 5 y3 - 3 y5 + 8 y2, 3 y1, 0, 0, 3 y6, 0, 3 y7, 3 y3, 3 y4, 3 y5, 3 y2]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
161 . Coloring, {4, 6, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, B, C, B, C, 1, 9]
B: [6, 8, 8, 6, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y3, 0, 0, 0, y1, 0, y5, y3, y4, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y9, y3, y4, 0, y5, 0, y6, y7, y8]


 
162 . Coloring, {4, 7, 8}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, A, A, A, B, B, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, 0, 0, y4, y5, y6]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y4, y2, y1, 0, y2 + y1 - y6 - y5 + y3, 0, y6, y5, 0, y4, y3]

  p' = s 6 - s 7   p = s 6 - s 8


 
163 . Coloring, {4, 7, 9}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, C, C, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 5 6 vs 7

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, 0, 0, y5, 0, y4]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y3, y1, y5, 0, y4, 0, y2, y3, 0, y6, 0]

  p = - s 2 + s 7


 
164 . Coloring, {4, 7, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2, y1, 0, 0, y2, 0, y3, 0, 0, y4, y5, y5]

  p' = - s 3 + s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -5 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 + 11 y6, 5 y1, 0, 5 y2, 0, 5 y3, 5 y4, 0, 5 y5, 5 y6]

  p = - s - s 2 + s 6 + s 7


 
165 . Coloring, {4, 7, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y3, 0, y2, 0, 0, y4, y5, y6]

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 + 13 y7, 3 y1, 3 y2, 0, 0, 3 y3, 0, 3 y4, 3 y5, 0, 3 y6, 3 y7]

  p = - s 2 - s 3 + s 7 + s 8


 
166 . Coloring, {4, 7, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y4, 0, 0, 0, 0, 0, y3, 0, -y4 + y3 - y1 + y2 + y5, y1, y2, y5]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y7, y3, y2, y1, y4, 0, y6, 0, 0, y5, y7]

  p = - s 3 + s 8


 
167 . Coloring, {4, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 6 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 2 y2, 0, -27 y2 + 16 y3 + 7 y1, 0, 0, 3 y1 - 7 y2 + 4 y3, 2 y3, -16 y2 + 10 y3 + 4 y1]

  p' = s 2 - s 5   p = - s + s 4   p' = - s + s 4

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y5, y6, y4, 0, y1, 0, y2, y3, y8, y9, y7]


 
168 . Coloring, {4, 8, 10}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, B, B, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, 0, 0, y5, y6, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1 + y5 + y2 - y4, y3, 0, y1, 0, y5, y2, y3, 0, y4]

  p = - s 5 + s 7   p = - s 5 + s 6


 
169 . Coloring, {4, 8, 11}

Ωp(Δ)=0:     p = s 2 + 56s 5 + 16s 6 - 64s 7 - 256s 8   p' = s 2 - 16s 4 - 8s 5 + 16s 6 + 64s 7   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 7, 7, 7, A, A, B, B, B, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 3 vs 6 5 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 5 y3 - y1 - y2, y3, 0, y1, 0, 0, y3, y2, y3]

  p = - s + s 4   p' = - s 2 + s 5   p' = - s + s 4

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y4, y1, y2 + y3 - y5, 0, 0, y1, 0, y2, y3, y4, 0, y5]

  p = - s 5 + s 7   p = - s 5 + s 6   p = - s 5 + s 8


 
170 . Coloring, {4, 8, 12}

Ωp(Δ)=0:     p = - s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8   p = s 2 + 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, B, B, B, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, 0, 0, 0, 0, y5, 0, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, y2, y1 + y3 - y5, y4, y1, y2, 0, y3, 0, y4, 0, y5]

  p = - s 3 + s 4 - s 5 + s 6   p' = - s 3 + s 7   p = - s 3 + s 7


 
171 . Coloring, {4, 9, 10}

Ωp(Δ)=0:     p = 3s 2 + 10s 3 + 40s 5 + 32s 6 + 32s 7 + 128s 8   p' = - 3s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, B, C, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, y6, 0, 0, y5, 0, y4, 0, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y6, y5, 0, y4, 0, y3, y2, y1, y8, y7]


 
172 . Coloring, {4, 9, 11}

Ωp(Δ)=0:     p = 3s 2 - 10s 3 - 40s 5 + 32s 6 - 32s 7 + 128s 8   p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, B, C, C, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 3 vs 6 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, y1, y2, 0, y3, 0, 0, y1, y2, y3]

  p' = - s + s 4   p' = - s 2 + s 5   p = - s + s 4

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, 0, y5, y6, y6, y7, 0]

  p = - s 3 + s 8


 
173 . Coloring, {4, 9, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 + 24s 5 - 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, B, C, C, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y1 + 2 y4 - y3, 0, 0, 0, 0, 0, y1, 0, -y2 + y4, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y2, y1, y3, y6, y4, 0, y5, 0, y6, y7, 0]

  p = - s 3 + s 8


 
174 . Coloring, {4, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, 0, y5, 0, 0, y4, y7, y6]

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-7 y1 - 7 y2 - 7 y3 + 9 y7 + 9 y4 - 7 y5 + 9 y6, 0, 7 y1, 0, 0, 7 y2, 0, 7 y3, 7 y7, 7 y4, 7 y5, 7 y6]

  p = - s 2 - s 3 + s 7 + s 8


 
175 . Coloring, {4, 10, 12}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 2 y3 - 2 y5, 0, 0, 0, 0, y2, 0, y3, 2 y5, y4, y5]

  p = s 3 - s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y8, y7, y6, y5, 0, y4, 0, y3, y2, y1]


 
176 . Coloring, {4, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, 0, y3, y4, y5, y6]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 2 y3 - 2 y7, y2, 0, y3, y4, 0, y5, 0, 2 y7, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
177 . Coloring, {5, 6, 7}

R: [7, 7, 7, 6, 3, 3, A, C, B, C, 1, 5]
B: [6, 8, 8, 7, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y5, y6, 0, 0, y4, y5, y7]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 0, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
178 . Coloring, {5, 6, 8}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, C, 1, 5]
B: [6, 8, 8, 7, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, 0, y6, 2 y4]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y7, 0, 2 y6, 0, y6, y5, y4, y2, y3, 0, y1]

  p = s 6 - s 8


 
179 . Coloring, {5, 6, 9}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, C, 1, 5]
B: [6, 8, 8, 7, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y4, y3, y5, 0, 0, 0, y6, 4 y3]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, y5, 2 y3, y6, y7, 0]

  p = - s 2 + s 8

 ยป SYNC'D !RANK'D


 
180 . Coloring, {5, 6, 10}

R: [7, 7, 7, 6, 3, 3, B, C, B, 2, 1, 5]
B: [6, 8, 8, 7, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y6, y3, 0, y4, y6, y2, 0, 0, 0, y5, y6]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y7, y2, 2 y7, y3, y4, y5, y6]

  p = s 6 - s 8


 
181 . Coloring, {5, 6, 11}

R: [7, 7, 7, 6, 3, 3, B, C, B, C, 4, 5]
B: [6, 8, 8, 7, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[11 y1 + 11 y2 + 11 y3 - 2 y4 - 2 y5 + 11 y6 - 39 y7, 2 y1, 0, 0, 0, 2 y2, 2 y3, 2 y4, 3 y1 + 3 y2 + 3 y3 + 3 y6 - 11 y7, 2 y5, 2 y6, 2 y7]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
182 . Coloring, {5, 6, 12}

R: [7, 7, 7, 6, 3, 3, B, C, B, C, 1, 9]
B: [6, 8, 8, 7, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y2, 0, y1, 0, 0, y4, y3, 0, 3 y1 - 4 y4, 0, y5, 3 y4]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y8, y4, y5, 0, y6, y7, y8]

  p = - s 3 + s 9


 
183 . Coloring, {5, 7, 8}

R: [7, 7, 7, 6, 3, A, A, B, B, C, 1, 5]
B: [6, 8, 8, 7, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, 0, 0, y2, 2 y4, y1]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 5 y6, 5 y7, 5 y3, 0, 5 y6 - 5 y7, 5 y4, 5 y5, 5 y2, 5 y6 - 5 y7, 5 y1, 5 y7 + 11 y3 + 11 y4 - 5 y5 - 5 y2 - 15 y6 + 11 y1]

  p' = s 4 + s 5 - s 7 - s 8   p = - s 4 + s 6 + s 7 - s 9   p = - s 4 + s 10


 
184 . Coloring, {5, 7, 9}

R: [7, 7, 7, 6, 3, A, A, C, C, C, 1, 5]
B: [6, 8, 8, 7, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y3, 0, y1, 0, y2, y3, y4, 0, 0, y5, 0, y6]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y1 - y4, y3, 0, y4, y2, y5, 2 y4, y4, y6, 0]

  p' = s 4 - s 7   p = s 4 - s 7   p' = s 5 - s 8


 
185 . Coloring, {5, 7, 10}

Ωp(Δ)=0:     p = - 2s 2 - s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, y6, y4, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, -3 y1 - 3 y4 - 3 y2 - 3 y3 + 10 y6 - 3 y5, 3 y1, 0, 3 y4, 3 y2, 3 y3, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
186 . Coloring, {5, 7, 11}

Ωp(Δ)=0:     p = 2s 2 - s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, C, 4, 5]
B: [6, 8, 8, 7, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y7, y8, y6, y5, y4, 0, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y8, 3 y7, 3 y6, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, 3 y4, -3 y8 - 3 y7 - 3 y6 - 3 y5 - 6 y4 - 3 y3 + 13 y2 + 13 y1, 3 y1]

  p = s 3 - s 5 - s 8 + s 10   p' = s 3 + s 4 - s 8 - s 9


 
187 . Coloring, {5, 7, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, C, 1, 9]
B: [6, 8, 8, 7, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2 - y3 - y4 + y5 + y6, 0, y1, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p' = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = s 2 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y3 - y8, y2, y3, y8, y4, y5, 0, y6, y7, y8]

  p' = - s 6 + s 9   p = - s 6 + s 9


 
188 . Coloring, {5, 8, 9}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, C, 1, 5]
B: [6, 8, 8, 7, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y10, y9, y8, 0, y7, y6, y5, y4, y3, y2, y1]


 
189 . Coloring, {5, 8, 10}

R: [7, 7, 7, 6, 3, A, B, B, B, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y4, y3, 0, -2 y3 + 4 y5, -y3 + 2 y5, y2, 0, 0, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 2 y2 - y4, 8 y2 - 4 y4 - 2 y1, 0, 4 y2 - 2 y4 - y1, y1, y2, y3, y4, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6   p = - s 4 + s 8


 
190 . Coloring, {5, 8, 11}

R: [7, 7, 7, 6, 3, A, B, B, B, C, 4, 5]
B: [6, 8, 8, 7, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4, y3, y2, y1, y4 - y3 - y2 + y1 - y7 + y6 + y5, 0, 0, y7, y6, y5]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[8 y1 - 4 y2 - 10 y3, y1, y2, 0, 0, y3, 4 y1 - 2 y2 - 5 y3, y5, 3 y1 - 4 y3 - y5 + y4, 10 y1 - 5 y2 - 12 y3, 0, y4]

  p' = s 7 - s 8   p' = s 6 - s 8   p' = s 5 - s 8   p = s 5 - s 9


 
191 . Coloring, {5, 8, 12}

R: [7, 7, 7, 6, 3, A, B, B, B, C, 1, 9]
B: [6, 8, 8, 7, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y2, y3, 0, y5, y6, y7, y4]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y2, 4 y2 - 2 y3, y5, 2 y2 - y3, y3, y4, 0, y7, 0, y6]

  p = - s 3 + s 8   p' = s 3 - s 8


 
192 . Coloring, {5, 9, 10}

R: [7, 7, 7, 6, 3, A, B, C, C, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 2 y2 - 3 y3, 0, 2 y5, y3, y4, 0, 0, y5, y6, 2 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
193 . Coloring, {5, 9, 11}

R: [7, 7, 7, 6, 3, A, B, C, C, C, 4, 5]
B: [6, 8, 8, 7, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y7 - y5 + y6 + y2 - y3 - y4, y1, y7, y5, y6, 0, 0, y2, y3, y4]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y8, 2 y5, y7, y6, 0]

  p = - s 4 + s 9


 
194 . Coloring, {5, 9, 12}

Ωp(Δ)=0:     p = s 2 - 3s 3 - 6s 4 - 8s 5 - 24s 6 - 32s 7 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, C, C, 1, 9]
B: [6, 8, 8, 7, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y2 + y3 + y4 - y5 + y6, 0, y1, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y2 - y3 - y4 + y5 + y6 - y7, y1, y2, 2 y3, y3, y4, y5, 0, y6, y7, 0]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


M              \ ;       N

$ [ [0, 0, 0, 0, 0, 270, 0, 0, 402, 168, 0, 892] , [0, 0, 0, 352, 0, 84, 0, 624, 0, 672, 0, 0] , [0, 0, 0, 168, 0, 336, 0, 210, 0, 576, 0, 442] , [0, 352, 168, 0, 402, 0, 270, 0, 0, 0, 540, 0] , [0, 0, 0, 402, 0, 446, 0, 221, 0, 663, 0, 0] , [270, 84, 336, 0, 446, 0, 260, 0, 0, 0, 336, 0] , [0, 0, 0, 270, 0, 260, 0, 260, 667, 681, 0, 1326] , [0, 624, 210, 0, 221, 0, 260, 0, 0, 0, 417, 0] , [402, 0, 0, 0, 0, 0, 667, 0, 0, 0, 663, 0] , [168, 672, 576, 0, 663, 0, 681, 0, 0, 0, 704, 0] , [0, 0, 0, 540, 0, 336, 0, 417, 663, 704, 0, 804] , [892, 0, 442, 0, 0, 0, 1326, 0, 0, 0, 804, 0] ] $     $ [ [0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1] , [0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1] , [0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0] , [0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0] , [0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0] , [1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0] , [1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0] , [0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0] ] $

τ= 72 , r'= 1/2

R: [7, 7, 7, 6, 3, A, B, C, C, C, 1, 9]
B: [6, 8, 8, 7, A, 3, A, B, B, 2, 4, 5]

Ranges

Action of R on ranges, [[24], [25], [25], [23], [20], [24], [25], [25], [20], [24], [25], [25], [23], [10], [21], [1], [12], [13], [13], [28], [3], [29], [29], [29], [27], [4], [4], [4], [2]]
Action of B on ranges, [[10], [21], [6], [17], [22], [11], [26], [7], [22], [11], [26], [7], [18], [24], [24], [15], [12], [28], [8], [12], [9], [28], [28], [8], [19], [16], [16], [5], [14]]

Cycles:    R , {{9, 12}, {1, 7, 11}},   B , {{2, 4, 7, 8, 10, 11}}

β({1, 6}) = 135/6928
β({1, 9}) = 201/6928
β({1, 10}) = 21/1732
β({1, 12}) = 223/3464
β({2, 4}) = 11/433
β({2, 6}) = 21/3464
β({2, 8}) = 39/866
β({2, 10}) = 21/433
β({3, 4}) = 21/1732
β({3, 6}) = 21/866
β({3, 8}) = 105/6928
β({3, 10}) = 18/433
β({3, 12}) = 221/6928
β({4, 5}) = 201/6928
β({4, 7}) = 135/6928
β({4, 11}) = 135/3464
β({5, 6}) = 223/6928
β({5, 8}) = 221/13856
β({5, 10}) = 663/13856
β({6, 7}) = 65/3464
β({6, 11}) = 21/866
β({7, 8}) = 65/3464
β({7, 9}) = 667/13856
β({7, 10}) = 681/13856
β({7, 12}) = 663/6928
β({8, 11}) = 417/13856
β({9, 11}) = 663/13856
β({10, 11}) = 22/433
β({11, 12}) = 201/3464

Partitions
α([{1, 2, 3, 5, 7, 11}, {4, 6, 8, 9, 10, 12}]) = 1/1

b1 = {1, 2, 3, 5, 7, 11} ` , ` b2 = {4, 6, 8, 9, 10, 12}

Action of R and B on the blocks of the partitions: = [1, 2] [2, 1]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {5, 9, 12}
Rank2
R,B [7, 7, 7, 6, 3, A, B, C, C, C, 1, 9], [6, 8, 8, 7, A, 3, A, B, B, 2, 4, 5]
π2 [0, 0, 0, 0, 270, 0, 0, 402, 168, 0, 892, 0, 352, 0, 84, 0, 624, 0, 672, 0, 0, 168, 0, 336, 0, 210, 0, 576, 0, 442, 402, 0, 270, 0, 0, 0, 540, 0, 446, 0, 221, 0, 663, 0, 0, 260, 0, 0, 0, 336, 0, 260, 667, 681, 0, 1326, 0, 0, 417, 0, 0, 663, 0, 704, 0, 804]
u2 [0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1] (dim 1)
wpp [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]

 

 
195 . Coloring, {5, 10, 11}

Ωp(Δ)=0:     p = - 6s 2 - s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y8, y7, y6, y5, y4, y3, 0, 0, y2, y1, -y8 - y7 + y6 + y5 - y4 + y3 + y2 - y1]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[7 y1, 0, 7 y3, 0, 0, 7 y2, 9 y1 + 9 y3 + 9 y2 + 9 y4 - 7 y5 - 7 y6 + 9 y8 - 7 y7, 7 y4, 7 y5, 7 y6, 7 y8, 7 y7]

  p = s 3 + s 4 - s 8 - s 9


 
196 . Coloring, {5, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y6, 0, 0, y6, y3, 0, y6 + y4, y4, y5, y6]

  p' = s 5 - s 8   p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
197 . Coloring, {5, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, C, 4, 9]
B: [6, 8, 8, 7, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y7 + y1 - y2 - y3 - y4 + y5 + y6, 0, y1, y2, 0, y3, y4, y5, y6]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y7, y8, 0, y9, y6, y7]

  p = - s 5 + s 10


 
198 . Coloring, {6, 7, 8}

R: [7, 7, 7, 6, A, 3, A, B, B, C, 1, 5]
B: [6, 8, 8, 7, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y1, 0, y1, 0, y2, y3, y4, 0, 0, y5, 2 y3, y6]

  p = - s 4 + s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 11 y1 - 10 y2 + 11 y3 - 5 y4 - 5 y5 - 5 y6 + 11 y8 - 5 y7, 5 y2, 5 y1, 0, 5 y2, 5 y3, 5 y4, 5 y5, 5 y6, 5 y8, 5 y7]

  p = - s 5 - s 6 + s 8 + s 9   p = s 5 - s 7 - s 8 + s 10


 
199 . Coloring, {6, 7, 9}

R: [7, 7, 7, 6, A, 3, A, C, C, C, 1, 5]
B: [6, 8, 8, 7, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y3, 0, y1, 0, y2, y3, y4, 0, 0, y5, 0, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y7, y1, 0, y7, y6, y5, 2 y7, y4, y3, 0]

  p' = s 5 - s 8   p = s 5 - s 8


 
200 . Coloring, {6, 7, 10}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 10 9 vs 10 5 vs 9 4 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y4, y3, y4 - y5, 0, y4, y5, y2, 0, 0, y1, y5, y5]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6   p = - s 3 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -3 y1 + 3 y2, 3 y3 + 3 y1, 0, -3 y1 + 3 y2, 3 y3, 3 y2, 7 y3 + 4 y2 - 3 y4, 3 y1, 3 y3 + 3 y2, 3 y4]

  p' = - s 3 + s 7   p = - s 3 + s 9   p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

 ยป SYNC'D !RANK'D


 
201 . Coloring, {6, 7, 11}

Ωp(Δ)=0:     p = 2s 2 - s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, C, 4, 5]
B: [6, 8, 8, 7, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 8 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y2 - 3 y3 + 5 y4 - 3 y5 + 8 y7, -3 y2 - 3 y1 + 8 y4 - 3 y6 + 5 y7, 3 y2, 0, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8   p' = - s 3 + s 9


 
202 . Coloring, {6, 7, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, C, 1, 9]
B: [6, 8, 8, 7, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y6 + y7 - y2 - y3 + y4 + y5, 0, y1, 0, 0, y6, y7, 0, y2, y3, y4, y5]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1 + 2 y5 - y6, y1, y2, y5 + y6, y5, y3, y4, 0, y6, y7, y5]

  p' = s 6 - s 9   p = - s 5 + s 8   p' = s 5 - s 8


 
203 . Coloring, {6, 8, 9}

Ωp(Δ)=0:     p = - 2s 2 + 3s 3 - 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, C, 1, 5]
B: [6, 8, 8, 7, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y1, 0, y2, 0, -2 y1 - 2 y2 + 5 y3 + 5 y5 - 4 y4, y3, 4 y1 + 4 y2 - 6 y3 - 6 y5 + 5 y4, 0, 0, 5 y1 + 5 y2 - 8 y3 - 8 y5 + 6 y4, y5, y4]

  p' = - s 4 + s 7   p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y9, y8, 0, y9, y7, y6, y5, y4, y3, y2]

  p = - s 2 + s 10


 
204 . Coloring, {6, 8, 10}

R: [7, 7, 7, 6, A, 3, B, B, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y3, y1, 0, 2 y5, y5, y6, 0, 0, 2 y1 - y5, y4, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, y1, -3 y1 + 2 y2, y2, y3, y4, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6   p = - s 4 + s 8


 
205 . Coloring, {6, 8, 11}

R: [7, 7, 7, 6, A, 3, B, B, B, C, 4, 5]
B: [6, 8, 8, 7, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, -5 y1 + 11 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 5 y1, 5 y2, 5 y3, 5 y4, 0, 0, 5 y5, 5 y6, 5 y7]

  p = s + s 2 + s 3 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y5, y4, y5, 0, 0, y3, y5, y2, y1, y6, 0, y7]

  p = s 6 - s 8   p' = s 6 - s 8


 
206 . Coloring, {6, 8, 12}

R: [7, 7, 7, 6, A, 3, B, B, B, C, 1, 9]
B: [6, 8, 8, 7, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, y4, y3, 0, y2, y4, y1, y5 + y4]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y2, 2 y4, y3, y4, y5, y6, 0, y7, 0, y8]

  p = - s 5 + s 9


 
207 . Coloring, {6, 9, 10}

R: [7, 7, 7, 6, A, 3, B, C, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y3, y4, y6, 0, 0, y5, y7, 2 y4]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y8, y5, 0, y8, y6, y7, y4, y2, y3, y1]

  p = - s 3 + s 9


 
208 . Coloring, {6, 9, 11}

R: [7, 7, 7, 6, A, 3, B, C, C, C, 4, 5]
B: [6, 8, 8, 7, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, -7 y1 + 9 y2 - 7 y3 - 7 y4 + 9 y5 - 7 y6 + 9 y7, 7 y1, 7 y2, 7 y3, 7 y4, 0, 0, 7 y5, 7 y6, 7 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y3, y6, 0, 0, y1, y6, y4, 2 y6, y5, y7, 0]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
209 . Coloring, {6, 9, 12}

R: [7, 7, 7, 6, A, 3, B, C, C, C, 1, 9]
B: [6, 8, 8, 7, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y3, 0, 3 y2, 0, 0, 3 y1, 3 y3 + 3 y2, 0, 7 y3 + 7 y2 - 3 y1 - 3 y4, 3 y1, 3 y3 + 3 y2 - 3 y1, 3 y4]

  p = - s 3 + s 5   p = - s 3 + s 7   p' = - s 3 + s 7   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y1, y2, 2 y4, y4, y5, y7, 0, y6, y8, 0]

  p = - s 3 + s 9


 
210 . Coloring, {6, 10, 11}

Ωp(Δ)=0:     p = - 2s 2 + 5s 3 + 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y7, y4, y5, y1, y2, 0, 0, y3, y9, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y3, 0, 0, y2, y3, y7, y4, y8, y5, y6]

  p = - s 7 + s 9


 
211 . Coloring, {6, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 5 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y5, y2, y2 - y4, 0, 0, y4, y1, 0, y2, y4, y3, y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y2, y1, y6, y7, y5, y4, 0, y3, y8, y9]


 
212 . Coloring, {6, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, C, 4, 9]
B: [6, 8, 8, 7, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y2, y1, 0, y4, y5, 0, y8, y3, y7, y6]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y6, 0, y5, y4, y7, y3, 0, y8, y9, y7]

  p = - s 4 + s 10


 
213 . Coloring, {7, 8, 9}

R: [7, 7, 7, 6, A, A, A, B, C, C, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y6, y3, 0, 0, y4, y6, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y6, y7, y8, 0, y6, y5, y3, y4, 0, y2, y1]

  p = - s 6 + s 9


 
214 . Coloring, {7, 8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y4, y5, 0, 0, 2 y1, y1, y2, 0, 0, y3, 2 y1, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 5 y6, 5 y7, 0, 5 y3, 5 y4, 5 y5, 5 y1, 0, 5 y2, -5 y6 + 11 y7 - 5 y3 + 11 y4 - 5 y5 - 5 y1 + 11 y2]

  p = s 4 + s 5 - s 7 - s 8

 ยป SYNC'D !RANK'D


 
215 . Coloring, {7, 8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, 2 y1, 2 y2, 2 y3, 3 y5, 0, 0, 2 y4, 2 y5, 2 y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 2 y4, y3, 0, 0, y2, y4, y5, y6, 0, y7, y8]

  p = s 7 - s 9


 
216 . Coloring, {7, 8, 12}

Ωp(Δ)=0:     p = 2s 2 + 5s 3 - 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 + y2 - y3 - y4 + y5 + y6, 0, 0, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 10 y3, 11 y1 - 5 y2 - 15 y3 + 11 y4 - 5 y5 + 11 y6 - 5 y7, 5 y1, 5 y2, 5 y3, 5 y4, 5 y5, 0, 0, 5 y6, 5 y7]

  p = s 2 - s 5 - s 6 + s 9   p' = s 2 + s 3 + s 4 - s 6 - s 7 - s 8


 
217 . Coloring, {7, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[2 y3, y1, 0, 0, y2, y3, y4, 0, 0, y5, 0, 2 y3]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, 2 y1 - 2 y3, 0, y6, 2 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
218 . Coloring, {7, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 6 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y1, 2 y2, 2 y3, 3 y1, 0, 0, 2 y4, 0, 2 y5]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 2 y4, y2, 0, 0, y3, y4, y5, 2 y4, 0, y6, 0]

  p = - s 2 + s 7   p' = - s 2 + s 7

 ยป SYNC'D !RANK'D


 
219 . Coloring, {7, 9, 12}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 4 vs 6 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1 - 2 y2 - 2 y3 + 2 y4, 0, 0, 0, 0, y1 - y2 - y3 + y4, y1, 0, y2, y3, 0, y4]

  p' = s 4 - s 5   p = s 4 - s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y4, y2, y3, 2 y4, y4, y5, y6, 0, 0, y1, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
220 . Coloring, {7, 10, 11}

R: [7, 7, 7, 6, A, A, A, C, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y4, y4, y5, y6, 0, 0, y2, y3, y3]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[-5 y1 - 5 y2 - 5 y3 - 5 y7 + 11 y4 - 5 y5 + 11 y6, 0, 5 y1, 0, 0, 5 y2, 5 y3, 5 y7, 5 y4, 0, 5 y5, 5 y6]

  p = - s 2 - s 3 + s 7 + s 8


 
221 . Coloring, {7, 10, 12}

R: [7, 7, 7, 6, A, A, A, C, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y5, y3, 0, y6, y7, y4, y5]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, 0, y7, 3 y4]

  p = - s 5 + s 8


 
222 . Coloring, {7, 11, 12}

R: [7, 7, 7, 6, A, A, A, C, B, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y3, y1, 0, -y2 + y3 + y1 - y4 + y5 + y6, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, y3, y4, y7, y5, 0, 0, y6, y7]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
223 . Coloring, {8, 9, 10}

R: [7, 7, 7, 6, A, A, B, B, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y7, y4, 0, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y4, 0, y5, y3, y2, y7, y8, y9, y6]


 
224 . Coloring, {8, 9, 11}

R: [7, 7, 7, 6, A, A, B, B, C, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y3, y2, y1, 0, 0, y10, y9, y8, y7, y6, y5, y4]


 
225 . Coloring, {8, 9, 12}

R: [7, 7, 7, 6, A, A, B, B, C, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y1 + 2 y5 + 2 y2 - y3, 0, 0, 0, 0, y5 + y2 - y4, y1, 0, y5, y2, y3, y4]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y3, y2, y4, y7, y5, y6, 0, y8, y7, y9]

  p = s 6 - s 10


 
226 . Coloring, {8, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y4, 0, y3, y1, y7, y2, 0, 0, y5, y6, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, 0, y1, 0, 0, -3 y2 + 2 y5, y2, y3, y4, y5, 0, y6]

  p = s 5 - s 7   p' = - s 5 + s 7


 
227 . Coloring, {8, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, 0, 2 y3, y5, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y3, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 4 + s 8


 
228 . Coloring, {8, 11, 12}

Ωp(Δ)=0:     p = - 2s 2 + s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, 0, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y4, y1, y2, 0, y3, -3 y4 + 2 y6, y4, y5, 0, y6, 0, y7]

  p = s 4 - s 8   p' = s 4 - s 8


 
229 . Coloring, {9, 10, 11}

Ωp(Δ)=0:     p = 9s 3 - 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y7, 0, y8, y4, y5, y6, 0, 0, y3, y1, y2]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y4, 0, y3, 0, 0, y1, y2, y7, y8, y9, y5, y6]


 
230 . Coloring, {9, 10, 12}

Ωp(Δ)=0:     p = 6s 2 - 5s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y1 + 2 y4, y1, 0, 0, 0, -y2 + 2 y4, y2, 0, y4, -y3 + 2 y4, y3, y4]

  p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8   p = - s 4 + s 5

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y6, y7, y8, y9, y1, y2, 0, y3, y4, y5]


 
231 . Coloring, {9, 11, 12}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, 0, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, y3, y1, 0, 2 y5, y4, y5, y6, 0, y7, y8, 0]

  p = - s 4 + s 9


 
232 . Coloring, {10, 11, 12}

R: [7, 7, 7, 6, A, A, B, C, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, 0, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y4, 0, y3, 0, y1, y2, y5, y6, 0, y7, y8, y9]


 
233 . Coloring, {2, 3, 4, 5}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7   p = 3s 2 + 14s 3 - 24s 5 + 32s 6 - 32s 7 - 128s 8

R: [7, 8, 8, 7, 3, A, B, C, B, C, 1, 5]
B: [6, 7, 7, 6, A, 3, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[7 y1, 0, 9 y1 - 7 y2 + 9 y3 - 7 y4 - 7 y5 + 9 y6 - 7 y7, 0, 7 y2, 0, 7 y3, 7 y4, 0, 7 y5, 7 y6, 7 y7]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y1, -3 y1 - 3 y4 - 3 y3 - 3 y2 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y4, 0, 3 y3, 3 y2, 0, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
234 . Coloring, {2, 3, 4, 6}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, B, C, B, C, 1, 5]
B: [6, 7, 7, 6, 3, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[7 y5 - 3 y2 - 3 y4, 0, -3 y1 + 3 y5, 0, 3 y1, 0, 3 y2, 3 y5 - 3 y3, 0, 3 y3, 3 y4, 3 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -6 y5 - 3 y3 - 3 y1 - 3 y7 + 13 y2 - 3 y6 + 13 y4, 3 y5, 3 y3, 0, 3 y1, 3 y7, 0, 3 y2, 3 y6, 3 y5, 3 y4]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9


 
235 . Coloring, {2, 3, 4, 7}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, 2 y5, 0, y6, y5, y4]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 3 y2, 3 y1, -3 y2 - 3 y1 - 3 y7 - 3 y6 + 13 y5 - 3 y4 + 13 y3, 0, 3 y7, 3 y6, 0, 3 y5, 0, 3 y4, 3 y3]

  p = s 2 + s 3 - s 7 - s 8


 
236 . Coloring, {2, 3, 4, 8}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, A, B, B, B, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 4 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[-y3 - y2 - y1 + 5 y4, 0, 0, 0, y4, 0, y3, y2, 0, y4, y1, y4]

  p' = - s 3 + s 6   p' = - s 2 + s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y4, -y4 + 2 y1, y3, 0, y2, -y3 + 2 y1, 0, y1, -y2 + 2 y1, 0, y1]

  p' = s 6 - s 7   p' = s 5 - s 7   p' = s 4 - s 7   p = s 4 - s 8


 
237 . Coloring, {2, 3, 4, 9}

Ωp(Δ)=0:     p = 3s 2 + 2s 3 + 8s 5 + 32s 7   p = - 9s 2 + 4s 4 - 24s 5 + 16s 6 - 96s 7 + 64s 8

R: [7, 8, 8, 7, A, A, B, C, C, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 8 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y3, 0, 0, 0, 5 y3 - y1 - y2 - y4, 0, y3, y1, 0, y2, y3, y4 ]

  p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y4, y5, 0, y6, y7, y8, 0]


 
238 . Coloring, {2, 3, 4, 10}

Ωp(Δ)=0:     p = s 3 + 8s 6 + 16s 7   p' = s 3 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, A, B, C, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[7 y1, 9 y1 - 7 y7 + 9 y6 - 7 y5 - 7 y4 + 9 y3 - 7 y2, 0, 0, 7 y7, 0, 7 y6, 7 y5, 0, 7 y4, 7 y3, 7 y2]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 7 + s 8


 
239 . Coloring, {2, 3, 4, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, B, C, B, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y1, 2 y2, 0, -7 y1 + 3 y2 + 4 y4, -27 y1 + 7 y2 - 2 y3 + 16 y4, 0, 2 y3, -16 y1 + 4 y2 + 10 y4, 2 y4]

  p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 0, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
240 . Coloring, {2, 3, 4, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, B, C, B, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y5, y3, y5, y6, y4]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, y4, y5, 0, 0, y6, y7, y7]

  p = s 5 - s 8   p' = s 5 - s 8


 
241 . Coloring, {2, 3, 5, 6}

R: [7, 8, 8, 6, 3, 3, B, C, B, C, 1, 5]
B: [6, 7, 7, 7, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 9 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[6 y1 + 6 y2 - 5 y3 + 6 y4 - 5 y5, 0, 5 y1 + 5 y2 + 5 y4 - 5 y6, 0, 5 y1, 5 y2, 5 y3, 5 y4, 0, 0, 5 y5, 5 y6]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8   p = - s 2 - s 4 + s 5 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 6 y5 - 3 y2 + 13 y3 - 3 y4 + 13 y6, 0, 3 y1, 0, 3 y5, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8


 
242 . Coloring, {2, 3, 5, 7}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, C, 1, 5]
B: [6, 7, 7, 7, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y8, y6, y5, 0, y4, y8, y7]

  p = - s 5 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1 + 3 y4, 3 y1, -6 y1 - 9 y4 - 3 y2 + 13 y3 - 3 y5 + 13 y6, 0, 3 y4, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = s 3 - s 5 - s 6 + s 8   p = - s 3 + s 9   p = - s 3 - s 4 + s 6 + s 7


 
243 . Coloring, {2, 3, 5, 8}

R: [7, 8, 8, 6, 3, A, B, B, B, C, 1, 5]
B: [6, 7, 7, 7, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, 0, y8, 0, y6, y7, y2, y3, 0, y4, y5, y9]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -y1 - 3 y2 - y3 + 6 y5 - y4, y1, 2 y2, 0, y2, y3, 0, y5, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
244 . Coloring, {2, 3, 5, 9}

R: [7, 8, 8, 6, 3, A, B, C, C, C, 1, 5]
B: [6, 7, 7, 7, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[5 y1, 0, 11 y1 - 5 y2 - 5 y7 + 11 y8 - 5 y5 - 5 y6 + 11 y3 - 5 y4, 0, 5 y2, 5 y7, 5 y8, 5 y5, 0, 5 y6, 5 y3, 5 y4]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y4, 0, y3, y5, 0, 2 y3, y6, 2 y2, 0]

  p = - s 4 + s 7   p' = s 4 - s 7


 
245 . Coloring, {2, 3, 5, 10}

Ωp(Δ)=0:     p = - 9s 3 + 2s 4 + 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[6 y1 + 6 y2 + 6 y6 + 6 y8 - 5 y4 - 5 y7, 5 y1, 5 y2, 0, 5 y1 + 5 y2 + 5 y6 - 5 y5 - 5 y3 + 5 y8, 5 y6, 5 y4, 5 y5, 0, 5 y3, 5 y7, 5 y8]

  p = - s 4 - s 6 + s 7 + s 9   p = - s 4 - s 5 - s 6 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y4, y1, 0, y4, y5, 0, -2 y1 + 3 y4 + y5 - y2 + y3, y2, y4, y3]

  p = - s 5 + s 7   p = - s 5 + s 8   p = - s 5 + s 6


 
246 . Coloring, {2, 3, 5, 11}

Ωp(Δ)=0:     p = 9s 3 + 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, C, 4, 5]
B: [6, 7, 7, 7, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y2 - y3 + y4 + y5 + y6 - y7 - y8, y1, y2, y3, y4, y5, 0, y6, y7, y8]

  p = s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 0, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
247 . Coloring, {2, 3, 5, 12}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, C, 1, 9]
B: [6, 7, 7, 7, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y6, 0, 0, y6, y7, y4, y5, -y6 + y4, y3, y2]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y2 - y5, y2, y2, y5, y3, 0, 0, y4, y5, y5]

  p = - s 3 + s 9   p' = - s 3 + s 6   p' = s 4 - s 7   p = - s 3 + s 6


 
248 . Coloring, {2, 3, 6, 7}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, C, 1, 5]
B: [6, 7, 7, 7, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, y5, 0, y4, y6 - y5, 3 y5 - 2 y6 + y3, y3, 0, y2, y6 - y5, y1]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 6 y2 - 3 y3 + 13 y5 - 3 y4 - 3 y7 + 13 y6, 3 y2, 3 y1, 0, 3 y2, 3 y3, 0, 3 y5, 3 y4, 3 y7, 3 y6]

  p = s 4 + s 5 - s 7 - s 8   p' = s 4 + s 5 - s 7 - s 8

 ยป SYNC'D !RANK'D


 
249 . Coloring, {2, 3, 6, 8}

R: [7, 8, 8, 6, A, 3, B, B, B, C, 1, 5]
B: [6, 7, 7, 7, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[3 y1, 0, 3 y3, 0, 3 y2, -7 y1 - 7 y3 + 14 y2 + 8 y5 - 3 y4, 3 y6, -7 y1 - 7 y3 + 8 y2 - 3 y6 + 14 y5, 0, 3 y5, 3 y4, -5 y1 - 5 y3 + 7 y2 + 7 y5]

  p' = - s 4 + s 7   p = s 4 - s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y3, y4, 2 y4, 0, y4, y2, 0, y1, -y3 - 4 y4 - y2 + 6 y1, 0, y1]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 2 - s 8   p' = s 2 - s 5


 
250 . Coloring, {2, 3, 6, 9}

R: [7, 8, 8, 6, A, 3, B, C, C, C, 1, 5]
B: [6, 7, 7, 7, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[3 y1, 0, 3 y2, 0, 3 y5, 3 y6, 3 y3, 3 y4, 0, 8 y1 - 7 y2 - 7 y5 + 14 y3 - 3 y4, 7 y1 - 5 y2 - 5 y5 + 7 y3, 14 y1 - 7 y2 - 7 y5 - 3 y6 + 8 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y3, y4, 0, 2 y3, y6, y5, 0]

  p = s 4 - s 7   p' = s 4 - s 7


 
251 . Coloring, {2, 3, 6, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[3 y2, 5 y2 - 3 y1 - 3 y3 - 3 y4 + 5 y5 - 3 y6 - 3 y7 + 5 y8 - 3 y9, 3 y1, 0, 3 y3, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8, 3 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y4, y4 + y1 - y3 - y2 + y5, 0, y4, y1, 0, y3, y2, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8


 
252 . Coloring, {2, 3, 6, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, C, 4, 5]
B: [6, 7, 7, 7, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y5, y6, y4, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 6 y4 - 3 y6 - 3 y7 + 13 y2 - 3 y3 + 13 y5, 3 y1, 3 y4, 0, 0, 3 y6, 3 y7, 0, 3 y2, 3 y3, 3 y4, 3 y5]

  p = s 4 + s 5 - s 7 - s 8   p' = - s 4 - s 5 + s 7 + s 8


 
253 . Coloring, {2, 3, 6, 12}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, C, 1, 9]
B: [6, 7, 7, 7, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, 0, y7, 0, 0, y6, y5, y4, y3, y6, y2, y1]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, y6, y4, 0, 0, y5, y6, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7   p' = - s 5 + s 8

 ยป SYNC'D !RANK'D


 
254 . Coloring, {2, 3, 7, 8}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y6, y7, 2 y6, 0, y5, y4, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 2 y1, -y3 - 3 y1 - y2 + 6 y5 - y4, y3, 0, y1, y2, 0, y5, 0, y4, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
255 . Coloring, {2, 3, 7, 9}

Ωp(Δ)=0:     p = - 2s 2 + s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y4, 0, 0, 0, y5, y4, y3, 2 y4, 0, y2, 0, y1]

  p = s 3 - s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y4, y1, y2, 0, y4, y3, 0, 2 y4, 0, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

 ยป SYNC'D !RANK'D


 
256 . Coloring, {2, 3, 7, 10}

R: [7, 8, 8, 6, A, A, A, C, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, y7, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, -5 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y6 + 11 y5, 5 y1, 0, 5 y2, 5 y3, 0, 5 y4, 0, 5 y6, 5 y5]

  p = - s 3 - s 4 + s 6 + s 7


 
257 . Coloring, {2, 3, 7, 11}

R: [7, 8, 8, 6, A, A, A, C, B, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y5, 2 y5, 0, y4, y5, y6]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y6 - 3 y3 + 13 y4 - 3 y5 + 13 y7, 3 y1, 3 y2, 0, 0, 3 y6, 3 y3, 0, 3 y4, 0, 3 y5, 3 y7]

  p = - s 2 - s 3 + s 7 + s 8


 
258 . Coloring, {2, 3, 7, 12}

R: [7, 8, 8, 6, A, A, A, C, B, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, y6, 2 y7, y5, y3, y4, y2]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y6, y2, y3, y4, y6, y1, 0, 0, 0, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
259 . Coloring, {2, 3, 8, 9}

R: [7, 8, 8, 6, A, A, B, B, C, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y4, 0, 0, 0, y3, y2, y1, 2 y2, 0, y4, y3 + y2, 2 y2 + y1]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y4, y1, 0, y6, y5, 0, y4, y3, y7, y6]

  p' = s 5 - s 8   p = s 5 - s 8


 
260 . Coloring, {2, 3, 8, 10}

Ωp(Δ)=0:     p = - 2s 2 - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y3, y3, y4, y5, 0, y6, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y2 + y3 - y4 - y1 + y5, 2 y2, 0, y2, y3, 0, y4, y1, 0, y5]

  p = - s 5 + s 6   p = - s 5 + s 7


 
261 . Coloring, {2, 3, 8, 11}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, -y2 - y1 - y3 - y4 + 6 y5 - y6, 0, 0, y3, y4, 0, y5, y6, 0, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
262 . Coloring, {2, 3, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, 2 y2, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, 2 y4, y5, y4, y3, 0, 0, y6, 0, 2 y4]

  p' = - s 4 + s 7   p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
263 . Coloring, {2, 3, 9, 10}

Ωp(Δ)=0:     p = 6s 2 + s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[5 y1, 11 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y7 - 5 y8, 0, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y3 + y1 + y2 - y4 - y5 + y6 + y7, y3, 0, y1, y2, 0, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
264 . Coloring, {2, 3, 9, 11}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, 0, y7, y8, y6, 0]


 
265 . Coloring, {2, 3, 9, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y1, 0, 0, 0, 0, 5 y2, 5 y3, 10 y2, 11 y1 - 15 y2 + 11 y3 - 5 y4 + 11 y5 - 5 y6, 5 y4, 5 y5, 5 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, 2 y1, 2 y5, 2 y4, -4 y5 + 6 y4, -2 y5 + 3 y4, 2 y3, 0, 0, 2 y2, -4 y5 + 6 y4, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
266 . Coloring, {2, 3, 10, 11}

R: [7, 8, 8, 6, A, A, B, C, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y4, 0, y2, y3, y1, y5, y6, 0, y8, y9, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 7 + s 8


 
267 . Coloring, {2, 3, 10, 12}

R: [7, 8, 8, 6, A, A, B, C, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, y2, 0, 0, 0, y8, y3, y4, y5, y6, y7, y9]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, y1, y2, y6, y3, 0, 0, y5, y6, y7]

  p = - s 3 + s 8


 
268 . Coloring, {2, 3, 11, 12}

R: [7, 8, 8, 6, A, A, B, C, B, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y4, 2 y4, y3, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y3, y4, 0, y2, y1, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
269 . Coloring, {2, 4, 5, 6}

R: [7, 8, 7, 7, 3, 3, B, C, B, C, 1, 5]
B: [6, 7, 8, 6, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, 0, y6, 0, y5, 0, y4, y3, 0, 0, y1, y2]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 3 y1, 0, -3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
270 . Coloring, {2, 4, 5, 7}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, C, 1, 5]
B: [6, 7, 8, 6, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y6, 0, y5, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 - 3 y8 + 13 y9, 3 y1, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 3 y9]

  p = - s 4 - s 5 + s 9 + s 10


 
271 . Coloring, {2, 4, 5, 8}

R: [7, 8, 7, 7, 3, A, B, B, B, C, 1, 5]
B: [6, 7, 8, 6, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 6 y2 + 6 y4 + 6 y5 - 5 y3 - 5 y6, -5 y1 + 5 y2 + 5 y4 + 5 y5 - 5 y7, 5 y1, 0, 5 y2, 5 y3, 5 y4, 5 y5, 5 y6, 0, 5 y7]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
272 . Coloring, {2, 4, 5, 9}

R: [7, 8, 7, 7, 3, A, B, C, C, C, 1, 5]
B: [6, 7, 8, 6, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y3, 0, y2, 0, y6, y5, 0, y5, y4, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, 3 y1, 3 y2, 5 y1 - 3 y2 - 3 y3 + 5 y4 - 3 y5 - 3 y6 + 5 y7 - 3 y8, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
273 . Coloring, {2, 4, 5, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y9, 0, y6, y7, 0, y8, y5, y4]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, 7 y7, 7 y6, 0, 7 y5, 7 y4, 7 y3, 7 y2, 9 y7 + 9 y6 + 9 y5 - 7 y4 + 9 y3 - 7 y2 + 9 y1 - 7 y8, 7 y1, 7 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
274 . Coloring, {2, 4, 5, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, C, 4, 5]
B: [6, 7, 8, 6, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y1, y5, 0, y6, y7, 0, y7, y4, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[-3 y1 - 3 y2 - 3 y4 + 7 y5 - 3 y7 + 7 y8, -3 y3 + 6 y5 - 3 y6 + 6 y8, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s - 2s 2 - 2s 3 - s 4 + s 6 + 2s 7 + 2s 8 + s 9   p' = - s - 2s 2 - 2s 3 - s 4 + s 6 + 2s 7 + 2s 8 + s 9


 
275 . Coloring, {2, 4, 5, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, C, 1, 9]
B: [6, 7, 8, 6, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, 0, y4, y5, y3, y5, y2, y1]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, 3 y7 + 3 y6 - 3 y5, 7 y7 + 7 y6 - 3 y1 - 3 y2 - 3 y3 - 3 y4, 3 y1, 3 y7, 3 y2, 3 y6, 3 y3, 0, 3 y7 + 3 y6, 3 y4, 3 y5]

  p' = s 4 - s 9   p' = s 3 - s 8   p = - s 3 + s 8


 
276 . Coloring, {2, 4, 6, 7}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, C, 1, 5]
B: [6, 7, 8, 6, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y6, 0, y4, y5, 0, y3, y5, y2]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, 2 y8, 2 y7, 2 y6, 0, 2 y5, 2 y4, 2 y3, 2 y2, -2 y6 - 2 y4 - 2 y3 - 39 y2 + 11 y8 + 11 y7 + 11 y5 + 11 y1, 2 y1, 3 y8 + 3 y7 + 3 y5 - 11 y2 + 3 y1]

  p' = - s 3 + s 9   p = s 3 - s 9


 
277 . Coloring, {2, 4, 6, 8}

R: [7, 8, 7, 7, A, 3, B, B, B, C, 1, 5]
B: [6, 7, 8, 6, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 5 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[9 y1 - 11 y4 + 7 y2 - 2 y3, 0, 2 y2, 0, -2 y1 + 2 y4 - 2 y2 + 2 y3, 0, 2 y1, 2 y2, 0, 7 y1 - 9 y4 + 7 y2 - 2 y3, 2 y3, 2 y4]

  p' = - s 2 + s 5   p = - s 2 + s 8   p = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 2 y1, -3 y1 + 6 y2 + 4 y5 - 3 y4 - 3 y3, -6 y1 + 12 y2 + 8 y5 - 6 y4 - 6 y3, 0, -6 y1 + 8 y2 + 12 y5 - 6 y4 - 6 y3, 2 y4, -3 y1 + 4 y2 + 6 y5 - 3 y4 - 3 y3, 2 y2, 2 y3, 0, 2 y5]

  p = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9   p' = - s 3 + s 6


 
278 . Coloring, {2, 4, 6, 9}

R: [7, 8, 7, 7, A, 3, B, C, C, C, 1, 5]
B: [6, 7, 8, 6, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y1 + y2 - y4, 0, -2 y1 - y2 + 2 y4 + y3, 0, -y3 + 2 y1, 0, y1, -2 y1 - y2 + 2 y4 + y3, 0, y2, y4, y3]

  p = - s 2 + s 8   p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y5, y4, y6, 2 y3, y7, y8, 0]

  p = s 6 - s 9


 
279 . Coloring, {2, 4, 6, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[3 y2, 3 y1, -3 y2 + 3 y4, 0, -3 y1 + 7 y4 - 3 y6 - 3 y5 - 3 y3, 0, 3 y4, 3 y6, 0, 3 y5, 3 y4, 3 y3]

  p' = s 3 - s 8   p' = s 2 - s 7   p = s 2 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y7, y1, 0, y1 - 2 y7 + y6 + y5 + y4 - y3 - y2, y7, y6, y5, y4, y3, y2]

  p' = s 7 - s 8   p = s 7 - s 9


 
280 . Coloring, {2, 4, 6, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, C, 4, 5]
B: [6, 7, 8, 6, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 3 y4, 3 y3, 3 y2, 0, 3 y4 + 3 y3, 3 y4, 0, 3 y1, 3 y4 + 3 y3, 4 y4 + 7 y3 - 3 y2 - 3 y1]

  p' = s 3 - s 6   p' = s 4 - s 7   p' = s 2 - s 5   p = s 2 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y5 - 3 y3 - 3 y4 + 13 y6 - 3 y8 - 3 y9 + 13 y7, 3 y1, 3 y2, 0, 0, 3 y5, 3 y3, 3 y4, 3 y6, 3 y8, 3 y9, 3 y7]

  p = - s 6 - s 7 + s 9 + s 10


 
281 . Coloring, {2, 4, 6, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, C, 1, 9]
B: [6, 7, 8, 6, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y2, y5, y4, y5, y6, y3]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, y7, y5, y6, 0, y10, y8, y9]


 
282 . Coloring, {2, 4, 7, 8}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, 0, y3, y2, 0, y7, y6, y5]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1, y3, y2, 0, y4, y5, y6, y7, 0, y8, y9]


 
283 . Coloring, {2, 4, 7, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, 0, y2, y3, 0, y4, 0, y5]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y6, y2, y1, 0, y3, y5, y4, y6, 0, y7, 0]

  p = s 3 - s 8


 
284 . Coloring, {2, 4, 7, 10}

R: [7, 8, 7, 7, A, A, A, C, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y8, 0, y6, y7, 0, y3, y4, y5]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, 5 y1, -5 y1 - 5 y6 - 5 y7 - 5 y4 + 11 y5 - 5 y3 + 11 y2, 0, 5 y6, 5 y7, 5 y4, 5 y5, 0, 5 y3, 5 y2]

  p = - s 2 - s 3 + s 7 + s 8


 
285 . Coloring, {2, 4, 7, 11}

R: [7, 8, 7, 7, A, A, A, C, B, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y3, 0, y4, y6, 0, y5, y6, y1]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[3 y5, 3 y6, 3 y7, 0, 0, -3 y5 - 3 y6 - 3 y7 - 3 y1 - 3 y2 + 13 y3 - 3 y8 + 13 y4, 3 y1, 3 y2, 3 y3, 0, 3 y8, 3 y4]

  p = - s 3 - s 4 + s 8 + s 9


 
286 . Coloring, {2, 4, 7, 12}

R: [7, 8, 7, 7, A, A, A, C, B, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y7, y1, y2, y3, y4, 2 y3 - 3 y7, y5, 0, 0, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
287 . Coloring, {2, 4, 8, 9}

R: [7, 8, 7, 7, A, A, B, B, C, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 10 y1 + 4 y3 + 4 y2 - 16 y4, 0, 2 y3, 2 y2, 0, 2 y4, 16 y1 + 7 y3 + 7 y2 - 27 y4, 4 y1 + 3 y3 + 3 y2 - 7 y4]

  p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 9, 11, 12}}   

See Matrix
 

[0, 5 y1, 11 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y7 - 5 y8 - 5 y9, 5 y2, 0, 5 y3, 5 y4, 5 y5, 5 y6, 5 y7, 5 y8, 5 y9]

  p = - s - s 2 - s 3 + s 8 + s 9 + s 10


 
288 . Coloring, {2, 4, 8, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, y5, 0, y6, y7, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y6, 2 y5, 0, -2 y5 + 2 y2, y5, y4, y3, y2, 0, y1]

  p' = s 5 - s 7   p = s 5 - s 7


 
289 . Coloring, {2, 4, 8, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 5 y4 - y1 - y2 - y3, y4, 0, y1, y2, 0, y4, y3, y4 ]

  p' = s 2 - s 5   p = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[5 y7, 5 y5, 5 y6, 0, 0, 5 y3, 5 y4, -5 y7 + 11 y5 - 5 y6 - 5 y3 + 11 y4 - 5 y1 + 11 y2 - 5 y8, 5 y1, 5 y2, 0, 5 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
290 . Coloring, {2, 4, 8, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, 2 y3, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 5 y1, 11 y1 - 5 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7 - 5 y8, 5 y2, 5 y3, 5 y4, 5 y5, 5 y6, 0, 5 y7, 0, 5 y8]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9


 
291 . Coloring, {2, 4, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[7 y1, 9 y1 - 7 y4 + 9 y5 - 7 y6 - 7 y7 + 9 y2 - 7 y3, 0, 0, 7 y4, 0, 7 y5, 7 y6, 0, 7 y7, 7 y2, 7 y3]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y9, y8, 0, y7, y5, y6, y4, y2, y3, y1]


 
292 . Coloring, {2, 4, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y1, 2 y2, 0, -16 y1 + 4 y2 + 10 y3 + 10 y4, 2 y3, 0, 2 y4, -7 y1 + 3 y2 + 4 y3 + 4 y4, -27 y1 + 7 y2 + 16 y3 + 16 y4]

  p' = s 2 - s 5   p' = s 3 - s 6   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[11 y1 - 5 y4 - 5 y2 + 11 y3 - 5 y6 - 5 y7 + 11 y5 - 5 y8, 5 y1, 5 y4, 0, 0, 5 y2, 5 y3, 5 y6, 5 y7, 5 y5, 5 y8, 0]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
293 . Coloring, {2, 4, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[7 y1, 0, 0, 0, 0, 0, 7 y5, 7 y3, 7 y4, 14 y3, 7 y2, 9 y1 + 9 y5 - 21 y3 - 7 y4 + 9 y2]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, 5 y1, 5 y6, 5 y4, 5 y5, 5 y3, 5 y2, 11 y1 - 5 y6 - 5 y4 - 5 y5 - 5 y3 + 11 y2 + 11 y7 - 5 y8, 0, 5 y7, 5 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
294 . Coloring, {2, 4, 10, 11}

R: [7, 8, 7, 7, A, A, B, C, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, y1 - y7 + y2 - y3 - y4 + y5 - y6, 0, y1, y7, 0, y2, y3, 0, y4, y5, y6]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[y1, 0, -y1 - y7 - y5 - y2 + 2 y4 + 2 y3, 0, 0, y7, y6, y5, y4, y3, y2, -y6 + y4 + y3]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
295 . Coloring, {2, 4, 10, 12}

R: [7, 8, 7, 7, A, A, B, C, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y9, y2, y3, y4, y5, 0, y6, y7, y8]


 
296 . Coloring, {2, 4, 11, 12}

R: [7, 8, 7, 7, A, A, B, C, B, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y3, y2, y5, 2 y2, y4, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[11 y1 - 5 y5 - 5 y6 - 5 y2 + 11 y3 - 5 y4 + 11 y7 - 5 y8 - 5 y9, 5 y1, 5 y5, 0, 5 y6, 5 y2, 5 y3, 5 y4, 0, 5 y7, 5 y8, 5 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10


 
297 . Coloring, {2, 5, 6, 7}

R: [7, 8, 7, 6, 3, 3, A, C, B, C, 1, 5]
B: [6, 7, 8, 7, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y4, y6, y3, y6, 0, y5, y6, y7]

  p = - s 3 + s 8   p' = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y7, 0, 3 y6, 0, 3 y5, 3 y4, 3 y5, 3 y3, 3 y2, -3 y7 - 3 y6 - 6 y5 - 3 y4 + 13 y3 - 3 y2 + 13 y1, 3 y1]

  p = - s 4 + s 6 + s 7 - s 9   p' = - s 4 - s 5 + s 7 + s 8


 
298 . Coloring, {2, 5, 6, 8}

R: [7, 8, 7, 6, 3, 3, B, B, B, C, 1, 5]
B: [6, 7, 8, 7, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y5, y4, y5, 0, 0, y6, 2 y5]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 4 y1, 0, -10 y1 - 10 y3 + 22 y2 - 10 y5 + 22 y4, 0, -5 y1 - 5 y3 + 11 y2 - 5 y5 + 11 y4, 4 y3, -5 y1 - 5 y3 + 11 y2 - 5 y5 + 11 y4, 4 y2, 4 y5, 0, 4 y4]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7   p = - s 2 + s 8


 
299 . Coloring, {2, 5, 6, 9}

R: [7, 8, 7, 6, 3, 3, B, C, C, C, 1, 5]
B: [6, 7, 8, 7, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y3, y5, y4, y5, 0, 0, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, 0, y2, 0, y6, y1, y6, 2 y6, y5, y4, 0]

  p' = s 4 - s 7   p = - s 4 + s 7


 
300 . Coloring, {2, 5, 6, 10}

R: [7, 8, 7, 6, 3, 3, B, C, B, 2, 1, 5]
B: [6, 7, 8, 7, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y4, y2, 0, y3, y4, y5, y6, 0, 0, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, y5, y6, y5, y4, y3, y2, y1]

  p = - s 6 + s 8


 
301 . Coloring, {2, 5, 6, 11}

R: [7, 8, 7, 6, 3, 3, B, C, B, C, 4, 5]
B: [6, 7, 8, 7, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y2, y3, y1, y6, y7, y8, 0, 0, y4, y5]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = s 5 + s 6 - s 8 - s 9


 
302 . Coloring, {2, 5, 6, 12}

R: [7, 8, 7, 6, 3, 3, B, C, B, C, 1, 9]
B: [6, 7, 8, 7, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, -y6 + y2, 0, 0, y6, y5, y6, y4, 0, y3, y2]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, 0, y4, y3, y6, y2, y6, 0, y1, y3 - y6, y6]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
303 . Coloring, {2, 5, 7, 8}

R: [7, 8, 7, 6, 3, A, A, B, B, C, 1, 5]
B: [6, 7, 8, 7, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y4, 0, y2, y1, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 7 y1 + 7 y5 + 7 y2 - 9 y3 - 9 y4 + 7 y6 - 9 y7, 7 y1 + 5 y5 + 7 y2 - 9 y3 - 9 y4 + 7 y6 - 9 y7, 2 y1, 0, 2 y5, 2 y2, 2 y3, 2 y4, 2 y5, 2 y6, 2 y7]

  p = s 4 - s 6 - s 7 + s 9   p = - s 4 + s 10   p = - s 4 - s 5 + s 7 + s 8


 
304 . Coloring, {2, 5, 7, 9}

R: [7, 8, 7, 6, 3, A, A, C, C, C, 1, 5]
B: [6, 7, 8, 7, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y4, 0, y1, 0, y2, y4, y3, y4, 0, y6, 0, y5]

  p = - s 2 + s 7   p' = - s 2 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y1 - y5, y2, 0, y5, y3, y4, 2 y5, y5, y6, 0]

  p' = - s 4 + s 7   p' = - s 5 + s 8   p = s 4 - s 7


 
305 . Coloring, {2, 5, 7, 10}

R: [7, 8, 7, 6, 3, A, A, C, B, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y5, y8, y4, y6, 0, y7, y8, y9]

  p = - s 3 + s 10

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y3, 3 y2, 0, 3 y4, 3 y1, -3 y3 - 3 y2 - 3 y4 - 3 y1 + 10 y6 - 3 y5, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
306 . Coloring, {2, 5, 7, 11}

R: [7, 8, 7, 6, 3, A, A, C, B, C, 4, 5]
B: [6, 7, 8, 7, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y6, y5, y7, y4, y2, y3, 0, y1, y3, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 - 3 y8 + 13 y9, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 3 y9]

  p = s 4 + s 5 - s 9 - s 10


 
307 . Coloring, {2, 5, 7, 12}

R: [7, 8, 7, 6, 3, A, A, C, B, C, 1, 9]
B: [6, 7, 8, 7, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y7 - y4 + y5, y2, 2 y7 - y4 + y5, y7, y3, y4, 0, y5, y6, y7]

  p' = s 6 - s 9   p = s 5 - s 8   p' = s 5 - s 8


 
308 . Coloring, {2, 5, 8, 9}

R: [7, 8, 7, 6, 3, A, B, B, C, C, 1, 5]
B: [6, 7, 8, 7, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, 0, y7, 0, y6, y5, y4, y5, 0, y3, y2, y1]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y10, y9, y8, 0, y7, y6, y5, y4, y3, y2, y1]


 
309 . Coloring, {2, 5, 8, 10}

R: [7, 8, 7, 6, 3, A, B, B, B, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, -y4 + 2 y6, 0, 2 y4, y4, y3, y7, 0, y6, y5, 0]

  p = - s 5 + s 8   p' = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y4, 0, y4, 2 y1, y5, y3, y4 + 2 y5, 0, y2]

  p' = - s 4 + s 6   p = - s 4 + s 6   p = - s 4 + s 8


 
310 . Coloring, {2, 5, 8, 11}

R: [7, 8, 7, 6, 3, A, B, B, B, C, 4, 5]
B: [6, 7, 8, 7, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, y3, 0, y2, y1, -y8 + y7 + y6 - y5 + y4 + y3 + y2 - y1]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[7 y8, 7 y7, 7 y6, 0, 0, 7 y5, 7 y4, 7 y3, 7 y2, 7 y1, 0, -7 y8 + 9 y7 - 7 y6 - 7 y5 + 9 y4 - 7 y3 - 7 y2 + 9 y1]

  p = s 5 + s 6 - s 8 - s 9


 
311 . Coloring, {2, 5, 8, 12}

R: [7, 8, 7, 6, 3, A, B, B, B, C, 1, 9]
B: [6, 7, 8, 7, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, 2 y4, y3, y4, y5, y6, 0, y7, 0, y8]

  p = s 6 - s 9


 
312 . Coloring, {2, 5, 9, 10}

R: [7, 8, 7, 6, 3, A, B, C, C, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y4, y3, y5, y9, y6, y7, y8]


 
313 . Coloring, {2, 5, 9, 11}

R: [7, 8, 7, 6, 3, A, B, C, C, C, 4, 5]
B: [6, 7, 8, 7, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y2, y2 - y1 + y3 - y4 - y5 - y6 + y7 + y8, y1, y3, y4, y5, 0, y6, y7, y8]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y6 + 9 y7 - 7 y8, 7 y1, 7 y2, 0, 0, 7 y3, 7 y4, 7 y5, 7 y6, 7 y7, 7 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
314 . Coloring, {2, 5, 9, 12}

R: [7, 8, 7, 6, 3, A, B, C, C, C, 1, 9]
B: [6, 7, 8, 7, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[5 y1, 0, 9 y1 + 9 y2 - 7 y3 - 7 y4 + 9 y5 - 7 y6, 0, 0, 9 y1 + 9 y2 - 7 y3 - 7 y4 + 9 y5 - 7 y6, 5 y2, 9 y1 + 9 y2 - 7 y3 - 7 y4 + 9 y5 - 7 y6, 5 y3, 5 y4, 5 y5, 5 y6]

  p = - s 3 + s 9   p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y1, y3, 2 y6, y6, y7, y8, 0, y4, y5, 0]

  p = - s 6 + s 9


 
315 . Coloring, {2, 5, 10, 11}

R: [7, 8, 7, 6, 3, A, B, C, B, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y5 + y6 + y7 - y8 - y9, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[-5 y1 - 5 y2 - 5 y3 + 6 y4 + 6 y5 - 5 y6, 0, 5 y1, 0, 0, 5 y2, 5 y4 + 5 y5 - 5 y7, 5 y3, 5 y4, 5 y5, 5 y6, 5 y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
316 . Coloring, {2, 5, 10, 12}

R: [7, 8, 7, 6, 3, A, B, C, B, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y9, y8, y7, 0, 0, y7, y6, y5, y4, y3, y2, y1]

  p = s 7 - s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y8, y9, y7]


 
317 . Coloring, {2, 5, 11, 12}

R: [7, 8, 7, 6, 3, A, B, C, B, C, 4, 9]
B: [6, 7, 8, 7, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1 - y3 - y5 - y6 + y7 + y4, 0, y1, y3, y2, y5, y6, y7, y4]

  p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[3 y8, 3 y7, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2, 0, 5 y8 - 3 y7 + 5 y6 - 3 y5 + 5 y4 - 3 y3 + 5 y2 + 5 y1 - 3 y9, 3 y1, 3 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10


 
318 . Coloring, {2, 6, 7, 8}

R: [7, 8, 7, 6, A, 3, A, B, B, C, 1, 5]
B: [6, 7, 8, 7, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, -y4 + y6, 0, y2, y4, y3, y4, 0, y5, y6, y7]

  p' = s 5 - s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 4 y4 + 7 y3 + 3 y6 - 3 y1 - 3 y2 - 3 y5, 3 y4 + 3 y3 - 3 y6, 3 y1, 0, 3 y4 + 3 y3 - 3 y6, 3 y2, 3 y4, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 4 + s 10   p' = - s 4 + s 7   p = - s 4 + s 7   p' = - s 5 + s 8


 
319 . Coloring, {2, 6, 7, 9}

R: [7, 8, 7, 6, A, 3, A, C, C, C, 1, 5]
B: [6, 7, 8, 7, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y6, 0, y3, 0, y4, y6, y5, y6, 0, y1, 0, y2]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y4, y2, 0, y4, y5, y6, 2 y4, y6, y3, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
320 . Coloring, {2, 6, 7, 10}

R: [7, 8, 7, 6, A, 3, A, C, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 4 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2 + y7, y1, y2, 0, y3, y7, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9   p' = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y4, -3 y4 + 3 y2 + 3 y3, 0, 3 y4, 3 y2, 3 y3, -3 y4 + 7 y2 + 4 y3 - 3 y1, 3 y3, 3 y2 + 3 y3, 3 y1]

  p = s 3 - s 5   p' = - s 3 + s 5   p' = - s 4 + s 6   p' = - s 3 + s 7   p' = - s 4 + s 8


 
321 . Coloring, {2, 6, 7, 11}

Ωp(Δ)=0:     p = 5s 4 + 8s 5 - 8s 7 - 16s 8

R: [7, 8, 7, 6, A, 3, A, C, B, C, 4, 5]
B: [6, 7, 8, 7, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y6, y7, y8, 0, y4, y8, y5]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y6, 3 y7, 3 y8, 0, 0, 3 y5, -3 y6 - 3 y1 + 5 y2 - 3 y3 + 8 y4, 3 y1, 3 y2, 3 y3, -3 y7 - 3 y8 - 3 y5 + 8 y2 + 5 y4, 3 y4]

  p' = s 3 - s 9   p = s 3 - s 9


 
322 . Coloring, {2, 6, 7, 12}

R: [7, 8, 7, 6, A, 3, A, C, B, C, 1, 9]
B: [6, 7, 8, 7, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y4, y3, y4, y5, y6, y7, y8]

  p = s 3 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2 + 2 y4 - y5, y2, y1, y4 + y5, y4, y7, y6, 0, y5, y3, y4]

  p = s 5 - s 8   p' = - s 5 + s 8   p' = - s 6 + s 9


 
323 . Coloring, {2, 6, 8, 9}

R: [7, 8, 7, 6, A, 3, B, B, C, C, 1, 5]
B: [6, 7, 8, 7, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y5, 0, y4, 0, -2 y5 - 2 y4 + 5 y3 + 5 y2 - 4 y1, y3, 4 y5 + 4 y4 - 7 y3 - 6 y2 + 5 y1, y3, 0, 5 y5 + 5 y4 - 8 y3 - 8 y2 + 6 y1, y2, y1]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y3, y2, 0, y3, y4, y5, y6, y7, y8, y9]

  p = - s 7 + s 10


 
324 . Coloring, {2, 6, 8, 10}

R: [7, 8, 7, 6, A, 3, B, B, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, y7, y5, 0, 2 y4, y4, y2, y3, 0, 2 y5 - y4, y1, 0]

  p' = - s 5 + s 8   p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, y1, 2 y4 + 2 y2 - 2 y3, y4 + y2 - y3, y4, y2, 0, y3]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8


 
325 . Coloring, {2, 6, 8, 11}

R: [7, 8, 7, 6, A, 3, B, B, B, C, 4, 5]
B: [6, 7, 8, 7, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 5 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -5 y1 + 11 y5 - 5 y2 - 5 y3 - 5 y4 + 11 y6 - 5 y7 + 11 y8, 5 y1, 5 y5, 5 y2, 5 y3, 5 y4, 0, 5 y6, 5 y7, 5 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[6 y2 - 12 y4 - 8 y3 + 6 y1 + 6 y5, y2, 3 y2 - 6 y4 - 4 y3 + 3 y1 + 3 y5, 0, 0, 3 y2 - 8 y4 - 2 y3 + 3 y1 + 3 y5, y1, 3 y2 - 7 y4 - 3 y3 + 3 y1 + 3 y5, y4, y5, 0, y3]

  p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9   p = - s 3 + s 6


 
326 . Coloring, {2, 6, 8, 12}

R: [7, 8, 7, 6, A, 3, B, B, B, C, 1, 9]
B: [6, 7, 8, 7, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, -y4 + y6, 0, 0, y4, y2, y4, y3, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 10 y4 - 9 y1 - 3 y2 + 10 y3 - 3 y5, 3 y4 + 3 y3 - 3 y6, 6 y1, 3 y4, 3 y1, 3 y2, 3 y3, 0, 3 y5, 0, 3 y6]

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7   p = s 2 - s 5 - s 6 + s 9


 
327 . Coloring, {2, 6, 9, 10}

R: [7, 8, 7, 6, A, 3, B, C, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[y1 - y2 + y3 - y4 - y5 + y6 + y7 - y8 + y9, y1, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, y2, y3, y4, y5, y6, y7]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
328 . Coloring, {2, 6, 9, 11}

R: [7, 8, 7, 6, A, 3, B, C, C, C, 4, 5]
B: [6, 7, 8, 7, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -y1 + y4 - y5 - y6 + y7 + y2 - y3 + y8, y1, y4, y5, y6, y7, 0, y2, y3, y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y3, y7, y8, 0]

  p = - s 6 + s 9


 
329 . Coloring, {2, 6, 9, 12}

R: [7, 8, 7, 6, A, 3, B, C, C, C, 1, 9]
B: [6, 7, 8, 7, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-y1 - y3 - y2 + 2 y5 - y4, 0, y1, 0, 0, y3, y2, y3, -2 y3 + y5, y3, y4, y5]

  p' = - s 5 + s 8   p = s 3 - s 6   p' = s 3 - s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y8, y7, y6, 2 y5, y5, y4, y3, 0, y2, y1, 0]

  p = s 6 - s 9


 
330 . Coloring, {2, 6, 10, 11}

R: [7, 8, 7, 6, A, 3, B, C, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 10 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, 6 y4 - 8 y5 + 8 y1 + 4 y2 - 12 y3, 8 y4 - 17 y5 + 15 y1 + 12 y2 - 22 y3, 2 y4, 2 y5, 12 y4 - 22 y5 + 18 y1 + 14 y2 - 28 y3, 2 y1, 2 y2, 0, 2 y3, -4 y4 + 12 y5 - 8 y1 - 8 y2 + 14 y3, 8 y4 - 15 y5 + 13 y1 + 8 y2 - 18 y3]

  p' = - s 2 + s 7   p' = - s 3 + s 8   p' = - s 4 + s 9   p = - s + s 6   p' = - s + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[3 y1 + y2 - y3 - y5 - y4 + y6 + y7, 0, y1, 0, 0, y2, 2 y1, y3, y5, y4, y6, y7]

  p = - s 7 + s 8   p = - s 7 + s 9


 
331 . Coloring, {2, 6, 10, 12}

R: [7, 8, 7, 6, A, 3, B, C, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 + y4, y2, 0, 0, y4, y5, y6, y3, y4, y8, y7]

  p = - s 6 + s 9   p' = - s 6 + s 9

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3, -y1 + y5 + y6, y1, y2, -y3 - y2 + y5 + y6, y5 + y6 - y4, 0, y4, y5, y6]

  p' = - s 2 + s 4 - s 6 + s 8   p' = - s 2 + s 3 - s 6 + s 7   p = s 2 - s 3 + s 6 - s 7


 
332 . Coloring, {2, 6, 11, 12}

R: [7, 8, 7, 6, A, 3, B, C, B, C, 4, 9]
B: [6, 7, 8, 7, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y6, y5, y6, y7, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y7, y4, y5, y6, 0, y8, y9, y10]


 
333 . Coloring, {2, 7, 8, 9}

R: [7, 8, 7, 6, A, A, A, B, C, C, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y3, y2, y3, 0, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y6, y1, y2, 0, y6, y5, y3, y4, 0, y8, y7]

  p = - s 6 + s 9


 
334 . Coloring, {2, 7, 8, 10}

R: [7, 8, 7, 6, A, A, A, B, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y3, y3, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -y1 - y2 - y3 + 5 y5 - y4, y5, 0, y1, y5, y2, y3, 0, y5, y4]

  p = s 4 - s 8   p' = s 5 - s 7   p' = s 4 - s 6


 
335 . Coloring, {2, 7, 8, 11}

R: [7, 8, 7, 6, A, A, A, B, B, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, y4, 2 y3, y3, 0, y7, y6, y5]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y6, y7, y8, 0, 0, y6 - y7 + y8 + y4 - y1 - y2 - y3 + y5, y4, y1, y2, 0, y3, y5]

  p = - s 8 + s 9


 
336 . Coloring, {2, 7, 8, 12}

R: [7, 8, 7, 6, A, A, A, B, B, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, y6, y7, y5, y3, y4, y2]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 2 y4, y1 + y4 + y3 - y6, 2 y1 + 2 y3 - y2 - y5, y1, y4, y2, y3, 0, 0, y5, y6]

  p = s 2 + s 4 - s 5 - s 7   p' = s 2 + s 4 - s 5 - s 7   p' = - s 3 - s 5 + s 6 + s 8


 
337 . Coloring, {2, 7, 9, 10}

R: [7, 8, 7, 6, A, A, A, C, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y3, y1, 0, 0, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, 2 y1 - 2 y3, 0, y6, 2 y3]

  p = s 4 - s 7   p' = s 4 - s 7


 
338 . Coloring, {2, 7, 9, 11}

R: [7, 8, 7, 6, A, A, A, C, C, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y3, y2, y1, 2 y3, y3, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y6, y5, 0, 0, y4, y3, y2, y6, 0, y7, 0]

  p = - s 3 + s 8


 
339 . Coloring, {2, 7, 9, 12}

R: [7, 8, 7, 6, A, A, A, C, C, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y5, 0, 0, 0, 0, y5, y4, y5, y3, y2, 0, y1]

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y3, y1, y2, 2 y3, y3, y4, y5, 0, 0, y6, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
340 . Coloring, {2, 7, 10, 11}

R: [7, 8, 7, 6, A, A, A, C, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y6, y5, y7, 2 y4, y2, 0, y3, y4, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-5 y1 - 5 y5 - 5 y2 - 5 y3 + 11 y4 - 5 y6 + 11 y7, 0, 5 y1, 0, 0, 5 y5, 5 y2, 5 y3, 5 y4, 0, 5 y6, 5 y7]

  p = - s 2 - s 3 + s 7 + s 8


 
341 . Coloring, {2, 7, 10, 12}

R: [7, 8, 7, 6, A, A, A, C, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y7, y4, y5, 0, 0, y6, 3 y7]

  p = - s 5 + s 8


 
342 . Coloring, {2, 7, 11, 12}

R: [7, 8, 7, 6, A, A, A, C, B, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, y6, y3, 2 y6 - 2 y7, y4, 0, 0, y5, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
343 . Coloring, {2, 8, 9, 10}

R: [7, 8, 7, 6, A, A, B, B, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, y5, 0, 0, y6, y7, y8, y1, 0, y2, y3, y7]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y1 - y3 + y2 - y4 - y5 - y6 + y8 + y7, 0, y3, y2, y4, y5, y6, y8, y7]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
344 . Coloring, {2, 8, 9, 11}

R: [7, 8, 7, 6, A, A, B, B, C, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y5, y6, 2 y4, y4, 0, y3, y1, y2]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {1, 3, 6, 8, 9, 11, 12}}   

See Matrix
 

[5 y4 - y1 - y2 - y3 - y5 - y6 - y7, y4, y1, 0, 0, y2, y4, y3, y5, y4, y6, y7]

  p' = - s + s 8   p' = - s 2 + s 9   p = - s + s 8


 
345 . Coloring, {2, 8, 9, 12}

R: [7, 8, 7, 6, A, A, B, B, C, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 10

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y2 + y3 + y4 - y1 - y5, 0, 0, 0, 0, y2 + y3 - y4, y1, y2 + y3 - y4, y2, y3, y5, y4]

  p' = s 4 - s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 2 y8, 2 y7, 2 y6, 2 y5, 7 y8 - 9 y7 + 7 y6 - 9 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y4, 2 y3, 0, 2 y2, 7 y8 - 9 y7 + 7 y6 - 9 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y1 ]

  p' = s 3 + s 4 + s 5 - s 7 - s 8 - s 9   p = s 3 - s 6 - s 7 + s 10


 
346 . Coloring, {2, 8, 10, 11}

R: [7, 8, 7, 6, A, A, B, B, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y6, y5, y6, y3, 0, y4, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-2 y2 + 2 y5, 0, y1, 0, 0, y2, -2 y2 + 2 y5, y3, y4, y5, 0, y6]

  p' = - s 5 + s 7   p = - s 5 + s 7


 
347 . Coloring, {2, 8, 10, 12}

R: [7, 8, 7, 6, A, A, B, B, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y3, 0, 0, 0, y2, y7, y6, 2 y2, y5, y4, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y1 + 3 y2 - y3 + y4 + y6 - y5, 2 y2, y1, y2, y3, y4, 0, y6, 0, y5]

  p' = - s 4 + s 5 - s 6 + s 7   p = s 4 - s 5 + s 6 - s 7


 
348 . Coloring, {2, 8, 11, 12}

R: [7, 8, 7, 6, A, A, B, B, B, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, 0, y1, 2 y3, y3, y4, y2, y5, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[y2, y3, y4, 0, y5, y6, y3, -y2 + 5 y3 - y4 - y5 - y6 - y1, 0, y3, 0, y1]

  p = - s 3 + s 7   p' = - s 4 + s 8   p' = - s 3 + s 7


 
349 . Coloring, {2, 9, 10, 11}

R: [7, 8, 7, 6, A, A, B, C, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, y4, y9, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
350 . Coloring, {2, 9, 10, 12}

R: [7, 8, 7, 6, A, A, B, C, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y4, 5 y4 - 2 y2 - 2 y5, 0, 0, 0, 2 y2, 2 y4, 2 y1, 5 y4 - 2 y1 - 2 y3, 2 y3, 2 y4, 2 y5]

  p' = - s 5 + s 8   p = s 5 - s 6   p' = - s 5 + s 7   p' = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y2 + y3 - y4 + y5 + y6 - y7 - y8, y1, y2, y3, y4, y5, 0, y6, y7, y8]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
351 . Coloring, {2, 9, 11, 12}

R: [7, 8, 7, 6, A, A, B, C, C, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 2 y7, y7, y3, y4, y5, y6]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[5 y5 - y1 - y2 - y3 - y4 - y6, y5, y1, 0, y2, y3, y5, y4, 0, y5, y6, 0]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8


 
352 . Coloring, {2, 10, 11, 12}

R: [7, 8, 7, 6, A, A, B, C, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y2, 0, y8, y3, y4, y5, y6, y7, y9]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y6, 0, y7, 0, y8, y9, y5, y1, 0, y2, y3, y4]


 
353 . Coloring, {3, 4, 5, 6}

R: [7, 7, 8, 7, 3, 3, B, C, B, C, 1, 5]
B: [6, 8, 7, 6, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[y1 + y2 - y3 + y4 - y5 + y6, 0, y1, 0, y2, 0, y3, y4, 0, 0, y5, y6]

  p = - s - s 2 - s 3 + s 5 + s 6 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, 3 y1, 0, -3 y2 + 8 y3 - 3 y7 + 5 y6, 0, -3 y1 - 3 y4 + 5 y3 - 3 y5 + 8 y6, 3 y4, 3 y2, 3 y3, 3 y7, 3 y5, 3 y6]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
354 . Coloring, {3, 4, 5, 7}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, C, 1, 5]
B: [6, 8, 7, 6, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, y3, 0, y5 - y4 + y3, y2, 0, y5 + y3 - y2, y1, y5 + y3 - y1]

  p' = s 6 - s 7   p' = s 5 - s 7   p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y6 - 3 y4 + 13 y5 - 3 y7 - 3 y8 + 13 y9, 3 y1, 3 y2, 0, 3 y3, 3 y6, 3 y4, 3 y5, 3 y7, 3 y8, 3 y9]

  p = - s 4 - s 5 + s 9 + s 10


 
355 . Coloring, {3, 4, 5, 8}

R: [7, 7, 8, 7, 3, A, B, B, B, C, 1, 5]
B: [6, 8, 7, 6, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y5, 0, y4, y3, 0, y8, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y3, y1, y2, 0, y4, y5, y6, y7, y8, 0, y9]


 
356 . Coloring, {3, 4, 5, 9}

R: [7, 7, 8, 7, 3, A, B, C, C, C, 1, 5]
B: [6, 8, 7, 6, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[7 y1, 0, 9 y1 - 7 y2 + 9 y3 - 7 y4 - 7 y5 + 9 y6 - 7 y7, 0, 7 y2, 0, 7 y3, 7 y4, 0, 7 y5, 7 y6, 7 y7]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y4, y6, y5, y7, y8, y9, 0]


 
357 . Coloring, {3, 4, 5, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[-5 y1 + 11 y2 + 11 y3 - 5 y4 + 11 y5 - 5 y6 - 5 y7 + 11 y8, 5 y1, 5 y2, 0, 5 y3, 0, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y9, y4, y5, y6, y7, y8]


 
358 . Coloring, {3, 4, 5, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, C, 4, 5]
B: [6, 8, 7, 6, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, y1 + y3 + y2 - y6, y5, y1, 0, y4, y3, 0, y2, -y5 + 2 y1 - y4 + 2 y3 + 2 y2, y6]

  p' = - s 2 - s 4 + s 5 + s 7   p = - s 2 - s 4 + s 5 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[-3 y1 - 3 y4 + 8 y5 - 3 y6 + 5 y8, -3 y2 - 3 y3 + 5 y5 - 3 y7 + 8 y8, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = s - s 9   p' = s - s 9


 
359 . Coloring, {3, 4, 5, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, C, 1, 9]
B: [6, 8, 7, 6, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y5, y6, y7, -y5 + y6 + y7 - y1 - y2 + y3 + y4 - y8 + y9, y1, y2, y3, 0, y4, y8, y9]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
360 . Coloring, {3, 4, 6, 7}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, C, 1, 5]
B: [6, 8, 7, 6, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, y4, 0, y5, 0, y3, y6 - y4, 0, y2, y4, y1]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, 2 y8, 2 y7, 2 y6, 0, 2 y5, 2 y4, 2 y3, 2 y2, 11 y8 + 11 y7 - 2 y6 + 11 y5 - 2 y4 - 2 y3 - 39 y2 + 11 y1, 2 y1, 3 y8 + 3 y7 + 3 y5 - 11 y2 + 3 y1]

  p' = - s 3 + s 9   p = - s 3 + s 9


 
361 . Coloring, {3, 4, 6, 8}

R: [7, 7, 8, 7, A, 3, B, B, B, C, 1, 5]
B: [6, 8, 7, 6, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[-2 y1 + 9 y2 + 9 y3 - 11 y5 - 2 y4, 0, 2 y1, 0, -2 y2 - 2 y3 + 2 y4 + 2 y5, 0, 2 y2, 2 y3, 0, 7 y2 + 7 y3 - 2 y4 - 9 y5, 2 y4, 2 y5]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y6, y5, 2 y5, 0, 2 y4, y4, y2, y3, y1, 0, y7]

  p = s 6 - s 8   p' = s 6 - s 8


 
362 . Coloring, {3, 4, 6, 9}

R: [7, 7, 8, 7, A, 3, B, C, C, C, 1, 5]
B: [6, 8, 7, 6, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[4 y4 + 4 y5 + 3 y1 + 3 y2 - 7 y3, 0, 2 y4, 0, 2 y5, 0, 10 y4 + 10 y5 + 4 y1 + 4 y2 - 16 y3, 2 y1, 0, 2 y2, 2 y3, 16 y4 + 16 y5 + 7 y1 + 7 y2 - 27 y3]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = s 4 - s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y8, y4, y5, 2 y2, y6, y7, 0]

  p = - s 3 + s 9


 
363 . Coloring, {3, 4, 6, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y9, y8, y7, 0, y6, 0, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y2, y6, y3, y4, y5, y6, y7]

  p' = s 6 - s 8   p = s 6 - s 8


 
364 . Coloring, {3, 4, 6, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, C, 4, 5]
B: [6, 8, 7, 6, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -y1 + y4, y2 + y5, y1, 0, y3, y2, 0, y5, y4, y3]

  p = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y8, 3 y7, 3 y6, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y8 - 3 y6 - 3 y3 + 8 y2 + 5 y1, -3 y7 - 3 y5 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = - s 3 + s 9   p = - s 3 + s 9


 
365 . Coloring, {3, 4, 6, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, C, 1, 9]
B: [6, 8, 7, 6, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y2 - y3 - y4 - y5 + y6 + y7 - y8 + y9, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

  p = s 5 - s 6 + s 7 - s 8 + s 9 - s 10


 
366 . Coloring, {3, 4, 7, 8}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y5, 0, 0, 0, y4, 0, y3, y2, 0, y1, y7, y6]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, 7 y4, 7 y5, 7 y6, 0, 9 y4 - 7 y5 - 7 y6 - 7 y1 + 9 y2 + 9 y3 - 7 y7 + 9 y8, 7 y1, 7 y2, 7 y3, 0, 7 y7, 7 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
367 . Coloring, {3, 4, 7, 9}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, 0, y2, y3, 0, y4, 0, y5]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y1, y2, 0, y3, y4, y5, y6, 0, y7, 0]

  p = - s 3 + s 8


 
368 . Coloring, {3, 4, 7, 10}

R: [7, 7, 8, 7, A, A, A, C, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6 + y4, y1, 0, 0, y2, 0, y3, y6, 0, y5, y6, y4]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, 0, 5 y1, -5 y1 - 5 y4 - 5 y5 - 5 y3 + 11 y2 - 5 y6 + 11 y7, 0, 5 y4, 5 y5, 5 y3, 5 y2, 0, 5 y6, 5 y7]

  p = s 2 + s 3 - s 7 - s 8


 
369 . Coloring, {3, 4, 7, 11}

R: [7, 7, 8, 7, A, A, A, C, B, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, 0, y3, y5, 0, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y3, -3 y3 - 3 y1 - 3 y2 - 3 y4 - 3 y5 + 13 y6 - 3 y7 + 13 y8, 3 y1, 0, 0, 3 y2, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
370 . Coloring, {3, 4, 7, 12}

R: [7, 7, 8, 7, A, A, A, C, B, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y7, y3, y4, y5, y6, y1, 2 y5 - 3 y7, 0, 0, y2, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
371 . Coloring, {3, 4, 8, 9}

R: [7, 7, 8, 7, A, A, B, B, C, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 2 y2, 0, 7 y1 - 27 y2 - 2 y3 + 16 y4, 2 y3, 0, 3 y1 - 7 y2 + 4 y4, 2 y4, 4 y1 - 16 y2 + 10 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[0, y1, y2, y3, 0, y6, y4, y5, y7, y8, y9, y10]


 
372 . Coloring, {3, 4, 8, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y5, 0, y3, y5, 0, y4, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8


 
373 . Coloring, {3, 4, 8, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y2, y4, 0, y1, -y2 - y1 + 5 y4 - y3, 0, y4, y3, y4]

  p' = s 2 - s 5   p' = s 3 - s 6   p = s 2 - s 5

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, y3, y2, 0, 0, y4, y5, y6, y8, y7, 0, y9]


 
374 . Coloring, {3, 4, 8, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y3, y2, y4, 2 y2, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, 0, y9]


 
375 . Coloring, {3, 4, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y5, y4, 0, y3, y1, y2, y9, y8, y7, y6]


 
376 . Coloring, {3, 4, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, 10 y1 - 16 y3 + 4 y4, 2 y1, 0, 4 y1 - 7 y3 + 3 y4, 16 y1 - 2 y2 - 27 y3 + 7 y4, 0, 2 y2, 2 y3, 2 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, y9, 0]


 
377 . Coloring, {3, 4, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[7 y1, 0, 0, 0, 0, 0, 7 y2, 7 y3, 9 y1 + 9 y2 - 21 y3 + 9 y4 - 7 y5, 14 y3, 7 y4, 7 y5]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y1, y2, y3, y4, y5, y8, y6, 0, y7, y9, 0]


 
378 . Coloring, {3, 4, 10, 11}

R: [7, 7, 8, 7, A, A, B, C, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5, 0, y4, y3, 0, y2, y1, 0, y8, y7, y6]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, 0, y3, y1, y2, y7, y8, y9, y6]


 
379 . Coloring, {3, 4, 10, 12}

R: [7, 7, 8, 7, A, A, B, C, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y4, 0, 0, 0, 0, y2, y5, y6, 2 y5, y3, y4]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
380 . Coloring, {3, 4, 11, 12}

R: [7, 7, 8, 7, A, A, B, C, B, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, y3, y4, 2 y3, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1 - y2 + y3 + y4 + y5 - y6 - y7 + y8 - y9, y1, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10


 
381 . Coloring, {3, 5, 6, 7}

R: [7, 7, 8, 6, 3, 3, A, C, B, C, 1, 5]
B: [6, 8, 7, 7, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y4 - y5 + y6, 0, y1, 0, y2, y7, y3, y4, 0, y5, y7, y6]

  p = s 5 - s 9   p' = s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 0, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9


 
382 . Coloring, {3, 5, 6, 8}

R: [7, 7, 8, 6, 3, 3, B, B, B, C, 1, 5]
B: [6, 8, 7, 7, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y3, y2, y4, y7, 0, 0, y6, 2 y2]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, 0, 2 y2, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8


 
383 . Coloring, {3, 5, 6, 9}

R: [7, 7, 8, 6, 3, 3, B, C, C, C, 1, 5]
B: [6, 8, 7, 7, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[y5, 0, y4, 0, y3, y2, y5, y1, 0, 0, y5, 5 y5 - y4 - y3 - y2 - y1]

  p' = - s 2 + s 6   p = - s 2 + s 6   p' = s 3 - s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y7, y3, y4, 2 y7, y5, y6, 0]

  p = - s 2 + s 8


 
384 . Coloring, {3, 5, 6, 10}

R: [7, 7, 8, 6, 3, 3, B, C, B, 2, 1, 5]
B: [6, 8, 7, 7, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[5 y7, -14 y7 + 18 y6 + 18 y5 - 14 y4 + 18 y3 - 14 y2 + 18 y1, 5 y6, 0, 5 y5, -7 y7 + 9 y6 + 9 y5 - 7 y4 + 9 y3 - 7 y2 + 9 y1, 5 y4, 5 y3, 0, 0, 5 y2, 5 y1]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8   p' = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y3, y2, y3, y4, y5, y6, y7]

  p = s 6 - s 8


 
385 . Coloring, {3, 5, 6, 11}

R: [7, 7, 8, 6, 3, 3, B, C, B, C, 4, 5]
B: [6, 8, 7, 7, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y8, y6, y7, y5, y4, y3, 0, 0, y1, y2]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y3 + 8 y4 - 3 y5 + 5 y7, -3 y1 - 3 y2 + 5 y4 - 3 y6 + 8 y7, 0, 0, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
386 . Coloring, {3, 5, 6, 12}

R: [7, 7, 8, 6, 3, 3, B, C, B, C, 1, 9]
B: [6, 8, 7, 7, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, 0, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y2 - y3 + y4 + y5 - y6, 0, y1, y2, y7, y3, y4, 0, y5, y6, y7]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
387 . Coloring, {3, 5, 7, 8}

R: [7, 7, 8, 6, 3, A, A, B, B, C, 1, 5]
B: [6, 8, 7, 7, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 10

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y6, 0, y5, 0, y4, y3, y6 - y5 + y4 - y3, y2, 0, y6 + y4 - y2, y1, y6 + y4 - y1]

  p' = - s 3 + s 4 - s 7 + s 8   p' = s 2 - s 3 + s 6 - s 7   p = s 2 - s 4 + s 6 - s 8

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, y4 + y5 - y2 + y3, y4 + y5 - y2, y4 + y5 + y2 + y3 - y6 - y1, 0, y3, y6, y4, y5, y3, y1, y2]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8   p = - s 4 + s 10


 
388 . Coloring, {3, 5, 7, 9}

R: [7, 7, 8, 6, 3, A, A, C, C, C, 1, 5]
B: [6, 8, 7, 7, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y1, 0, -y1 + y3 + y4 - y2, 0, -2 y1 + y3 + y4, y1, y2, y3, 0, y4, 0, y3 + y4]

  p' = - s 4 + s 5   p = s 4 - s 5   p' = - s 4 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2 + y6, y2, y1, 0, y6, y3, y4, 2 y6, y6, y5, 0]

  p' = - s 4 + s 7   p' = - s 5 + s 8   p = s 4 - s 7


 
389 . Coloring, {3, 5, 7, 10}

R: [7, 7, 8, 6, 3, A, A, C, B, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 5 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[-5 y1 + 11 y2 + 11 y3 - 10 y7 - 5 y4 + 11 y5 - 5 y6 + 11 y8, 5 y1, 5 y2, 0, 5 y3, 5 y7, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8 ]

  p = s 3 - s 6 - s 7 + s 10   p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -3 y1 - 3 y2 - 6 y3 + 10 y5 - 3 y4, 3 y1, 0, 3 y3, 3 y2, 3 y3, -3 y3 + 3 y5, 3 y3, 3 y4, 3 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9


 
390 . Coloring, {3, 5, 7, 11}

R: [7, 7, 8, 6, 3, A, A, C, B, C, 4, 5]
B: [6, 8, 7, 7, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, -y4 + y5 - y2, -y1 + y4 + y5, y1, y2, 2 y4, -y3 + y4 + y5, 0, y3, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 8   p = - s 5 + s 9   p = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y9, 3 y8, 3 y7, 0, 0, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2, -3 y9 - 3 y8 - 3 y7 - 3 y6 - 3 y5 - 3 y4 + 13 y3 - 3 y2 + 13 y1, 3 y1]

  p = - s 4 - s 5 + s 9 + s 10


 
391 . Coloring, {3, 5, 7, 12}

R: [7, 7, 8, 6, 3, A, A, C, B, C, 1, 9]
B: [6, 8, 7, 7, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[2 y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y1, y2, y3, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y3 - y8, y2, y3, y8, y4, y5, 0, y6, y7, y8]

  p = - s 6 + s 9   p' = - s 6 + s 9


 
392 . Coloring, {3, 5, 8, 9}

R: [7, 7, 8, 6, 3, A, B, B, C, C, 1, 5]
B: [6, 8, 7, 7, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, 0, y2, 0, y7, y3, y4, y5, 0, y6, y9, y8]

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, y7, y8, y9, y10]


 
393 . Coloring, {3, 5, 8, 10}

R: [7, 7, 8, 6, 3, A, B, B, B, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, -5 y6 + 4 y1 - 2 y4, 0, 2 y6, y6, y5, y4, 0, 2 y1 - 2 y6 - y4, y3, 0]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y6, 2 y5, 0, y5, y4, y5, y3, y2, 0, y1]

  p' = s 5 - s 7   p = s 5 - s 7


 
394 . Coloring, {3, 5, 8, 11}

R: [7, 7, 8, 6, 3, A, B, B, B, C, 4, 5]
B: [6, 8, 7, 7, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3, y3 - y1 + y2 - y7 - y8 - y6 + y5 + y4, y1, y2, y7, y8, 0, y6, y5, y4]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y8, y9, y4, y5, y6, 0, y7]


 
395 . Coloring, {3, 5, 8, 12}

R: [7, 7, 8, 6, 3, A, B, B, B, C, 1, 9]
B: [6, 8, 7, 7, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y7, 0, 0, y7, y6, y5, y4, y5, y3, y2]

  p' = s 5 - s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y7, y4, 2 y6, y5, y6, y1, y2, 0, y3, 0, y8]

  p = - s 4 + s 9


 
396 . Coloring, {3, 5, 9, 10}

R: [7, 7, 8, 6, 3, A, B, C, C, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[-3 y1 + 10 y8 - 3 y7 - 3 y5 + 10 y6 - 3 y3 - 3 y4, 3 y1, 3 y8 + 3 y6 - 3 y2, 0, 3 y8, 3 y7, 3 y5, 3 y6, 0, 3 y3, 3 y4, 3 y2]

  p' = - s 4 - s 6 + s 7 + s 9   p = s 4 + s 6 - s 7 - s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y4, y3, y4, y7, y8, y6, y5]

  p = - s 3 + s 9


 
397 . Coloring, {3, 5, 9, 11}

R: [7, 7, 8, 6, 3, A, B, C, C, C, 4, 5]
B: [6, 8, 7, 7, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y2, y2 - y1 + y4 - y5 - y6 - y3 + y7 + y8, y1, y4, y5, y6, 0, y3, y7, y8]

  p = s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y2, y3, y1, 0, 0, y6, y4, y5, y9, y7, y8, 0]


 
398 . Coloring, {3, 5, 9, 12}

R: [7, 7, 8, 6, 3, A, B, C, C, C, 1, 9]
B: [6, 8, 7, 7, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y4, 0, y1, 0, 0, y1, y4, y3, 5 y4 - 2 y1 - 2 y3 - y2, y3, y4, y2]

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7   p' = - s 3 + s 7   p = - s 3 + s 9

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y1, y2, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = - s 3 + s 9


 
399 . Coloring, {3, 5, 10, 11}

R: [7, 7, 8, 6, 3, A, B, C, B, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3, -8 y7 + 5 y6 - 3 y5 + 5 y3 + 5 y2 - 8 y1, 0, 3 y2, -3 y8 - 11 y7 + 8 y6 - 3 y4 + 8 y3 + 8 y2 - 11 y1, 3 y1]

  p = - s - s 3 + s 7 + s 9   p' = - s - s 3 + s 7 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y9, y7, y8]


 
400 . Coloring, {3, 5, 10, 12}

R: [7, 7, 8, 6, 3, A, B, C, B, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 + y7, y2, 0, 0, y2, y3, y5, y4, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8   p' = - s 6 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y6, y1, y2, y4, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
401 . Coloring, {3, 5, 11, 12}

R: [7, 7, 8, 6, 3, A, B, C, B, C, 4, 9]
B: [6, 8, 7, 7, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y2 + y1 + y4 + y3 + y5 - y7 - y6, 2 y2, y4, y3, y5, y7, y6]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, -y1 + y2 + y4 + y5 + y3 - y8 - y9 + y6 - y7, 0, y4, y5, y3, y8, 0, y9, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
402 . Coloring, {3, 6, 7, 8}

R: [7, 7, 8, 6, A, 3, A, B, B, C, 1, 5]
B: [6, 8, 7, 7, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y8, 0, y9, 0, y1, y2, y3, y4, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 2 y8, 7 y8 - 9 y7 - 9 y6 + 7 y5 + 7 y4 + 7 y3 - 9 y2 + 7 y1, 2 y7, 0, 7 y8 - 9 y7 - 9 y6 + 7 y5 + 7 y4 + 7 y3 - 9 y2 + 7 y1, 2 y6, 2 y5, 2 y4, 2 y3, 2 y2, 2 y1]

  p' = s 5 + s 6 - s 8 - s 9   p = s 5 - s 7 - s 8 + s 10


 
403 . Coloring, {3, 6, 7, 9}

R: [7, 7, 8, 6, A, 3, A, C, C, C, 1, 5]
B: [6, 8, 7, 7, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y1, 0, y3, 0, y2, y1, 2 y3, y6, 0, y5, 0, y4]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y3, y7, y4, 2 y3, y5, y6, 0]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
404 . Coloring, {3, 6, 7, 10}

R: [7, 7, 8, 6, A, 3, A, C, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 4 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y2 - y7, 0, y3, y7, y4, y5, 0, y6, y7, y8]

  p' = - s 6 + s 9   p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 7 y4 - 3 y1 - 3 y2 - 3 y3, -4 y4 + 3 y1 + 3 y2 + 3 y3, 0, 7 y4 - 3 y1 - 3 y2 - 3 y3, -4 y4 + 3 y1 + 3 y2 + 3 y3, 7 y4 - 3 y1 - 3 y2 - 3 y3, 3 y1, 3 y2, 3 y4, 3 y3]

  p' = s 5 - s 7   p' = s 3 - s 7   p' = s 4 - s 8   p = s 3 - s 9   p' = s 6 - s 8


 
405 . Coloring, {3, 6, 7, 11}

R: [7, 7, 8, 6, A, 3, A, C, B, C, 4, 5]
B: [6, 8, 7, 7, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y8, y7, y6, y5, 2 y2, y4, 0, y3, y2, y1]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y3 - 3 y4 + 5 y5 - 3 y7 + 8 y8, -3 y1 - 3 y2 + 8 y5 - 3 y6 + 5 y8, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y7, 3 y6, 3 y8]

  p = - s 3 + s 9   p' = - s 3 + s 9


 
406 . Coloring, {3, 6, 7, 12}

R: [7, 7, 8, 6, A, 3, A, C, B, C, 1, 9]
B: [6, 8, 7, 7, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y9, y6, y7, y8, y5, y3, y4]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1 + 2 y7 - y5, y1, y2, y7 + y5, y7, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8   p' = - s 6 + s 9


 
407 . Coloring, {3, 6, 8, 9}

R: [7, 7, 8, 6, A, 3, B, B, C, C, 1, 5]
B: [6, 8, 7, 7, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y5, 0, y4, 0, y3, -6 y5 - 6 y4 + 5 y3 + 4 y2 + 4 y1 - y6, y2, y1, 0, 5 y5 + 5 y4 - 4 y3 - 2 y2 - 2 y1, y6, -8 y5 - 8 y4 + 6 y3 + 5 y2 + 5 y1]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y4, y2, 0, y4, y3, y5, y6, y7, y8, y9]

  p = - s 2 + s 10


 
408 . Coloring, {3, 6, 8, 10}

R: [7, 7, 8, 6, A, 3, B, B, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 - y5 + 2 y4, y2, 0, 4 y2 - 2 y5, 2 y2 - y5, y3, y4, 0, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, y5, y4, y5, y2, y3, 0, y1]

  p = - s 4 + s 8   p' = - s 4 + s 6   p = - s 4 + s 6


 
409 . Coloring, {3, 6, 8, 11}

R: [7, 7, 8, 6, A, 3, B, B, B, C, 4, 5]
B: [6, 8, 7, 7, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -5 y6 + 11 y4 - 5 y5 - 5 y3 - 5 y2 + 11 y1 - 5 y7 + 11 y8, 5 y6, 5 y4, 5 y5, 5 y3, 5 y2, 0, 5 y1, 5 y7, 5 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y1, y7, y1, 0, 0, -3 y1 + 2 y2, y2, y3, y4, y5, 0, y6]

  p' = - s 6 + s 8   p = - s 6 + s 8


 
410 . Coloring, {3, 6, 8, 12}

R: [7, 7, 8, 6, A, 3, B, B, B, C, 1, 9]
B: [6, 8, 7, 7, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, -y4 + y5, 0, 0, y4, y2, y3 + y4 - y5, y3, y4, y6, y5]

  p' = - s 4 + s 7   p = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y8, y7, 2 y5, y6, y5, y4, y3, 0, y2, 0, y1]

  p = s 2 - s 9


 
411 . Coloring, {3, 6, 9, 10}

R: [7, 7, 8, 6, A, 3, B, C, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y2, y1, y6, 0, y5, y4, y8, y3, 0, y7, y10, y9]

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y2, 0, y3, y1, y3, y7, y5, y6, y4]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
412 . Coloring, {3, 6, 9, 11}

R: [7, 7, 8, 6, A, 3, B, C, C, C, 4, 5]
B: [6, 8, 7, 7, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y2, y1, y8, y3, y4, y5, 0, y6, y7, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y3, y7, y8, 0]

  p = - s 3 + s 9


 
413 . Coloring, {3, 6, 9, 12}

R: [7, 7, 8, 6, A, 3, B, C, C, C, 1, 9]
B: [6, 8, 7, 7, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y3, 0, 5 y3 - 2 y4, 0, 0, 2 y2, 2 y3, 5 y3 - 4 y2 - 2 y1, 2 y1, 2 y2, 2 y3, 2 y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8   p = - s 4 + s 9

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = - s 3 + s 9


 
414 . Coloring, {3, 6, 10, 11}

R: [7, 7, 8, 6, A, 3, B, C, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, y4, y5, y6, y1, y2, y3, y7, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y4, y2 + y6, y2, y3, y5, y6, y7]

  p = - s 6 + s 8   p' = - s 6 + s 8


 
415 . Coloring, {3, 6, 10, 12}

R: [7, 7, 8, 6, A, 3, B, C, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 + y6, y2, 0, 0, y6, y4, y3, y8, y6, y5, y7]

  p = s 6 - s 9   p' = - s 6 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, y5, 0, y3, y2, y1]

  p = s 4 - s 9


 
416 . Coloring, {3, 6, 11, 12}

R: [7, 7, 8, 6, A, 3, B, C, B, C, 4, 9]
B: [6, 8, 7, 7, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y8, y7, 0, y6, 2 y3, y4, y5, y3, y2, y1]

  p = - s 2 + s 9

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 - y2 + y3 + y4 + y5 - y6 - y7 + y8 - y9, y1, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
417 . Coloring, {3, 7, 8, 9}

R: [7, 7, 8, 6, A, A, A, B, C, C, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y7, y6, y7, 0, y3, y4, y5]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y3, y1, y2, 0, y3, y4, 2 y1 - 3 y3, y5, 0, y6, y7]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
418 . Coloring, {3, 7, 8, 10}

R: [7, 7, 8, 6, A, A, A, B, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y5, y5, y3, y5, 0, y4, y6, 0]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 2 y6, 2 y5, 0, 7 y6 + 7 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y4, 7 y6 + 7 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y3, 0, 2 y2, 2 y1]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 - s 5 - s 6 + s 8


 
419 . Coloring, {3, 7, 8, 11}

R: [7, 7, 8, 6, A, A, A, B, B, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y6, y5, 2 y4, y4, 0, y3, y2, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[9 y1 - 7 y2 - 7 y3 - 7 y4 + 9 y5 + 9 y6 - 7 y7 + 9 y8, 7 y1, 7 y2, 0, 0, 7 y3, 7 y4, 7 y5, 7 y6, 0, 7 y7, 7 y8]

  p = s 3 + s 4 - s 8 - s 9


 
420 . Coloring, {3, 7, 8, 12}

R: [7, 7, 8, 6, A, A, A, B, B, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y1, y4, y3, y2, y1, y5, y6, 0, 0, y8, y7]

  p = - s 6 + s 9


 
421 . Coloring, {3, 7, 9, 10}

R: [7, 7, 8, 6, A, A, A, C, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y4, y2, 0, 0, y1, y4, y3, y4, 0, y5, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, y1, y2, 0, y4, y3, y4, 2 y1 - 2 y4, 0, y5, 2 y4]

  p' = - s 3 + s 6   p' = s 4 - s 7   p = - s 3 + s 6


 
422 . Coloring, {3, 7, 9, 11}

R: [7, 7, 8, 6, A, A, A, C, C, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y2, y3, y4, 2 y2, y2, 0, y1, 0, y5]

  p = - s 3 + s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y3, y4, y5, 0, 0, y6, y7, y1, y4, 0, y2, 0]

  p = - s 3 + s 8


 
423 . Coloring, {3, 7, 9, 12}

R: [7, 7, 8, 6, A, A, A, C, C, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, 0, 0, 0, y2, y1, y2, y3, y4, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 4 y1 - 6 y4, y1, y2, 4 y1 - 6 y4, 2 y1 - 3 y4, y3, y4, 0, 0, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
424 . Coloring, {3, 7, 10, 11}

R: [7, 7, 8, 6, A, A, A, C, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y5 + y6, 2 y5 - y6 + y2, y2, y3, y5, 0, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[-5 y1 - 5 y4 - 5 y5 - 5 y6 + 11 y2 - 5 y3 + 11 y7, 0, 5 y1, 0, 0, 5 y4, 5 y5, 5 y6, 5 y2, 0, 5 y3, 5 y7]

  p = s 2 + s 3 - s 7 - s 8


 
425 . Coloring, {3, 7, 10, 12}

R: [7, 7, 8, 6, A, A, A, C, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y4, y3, y4, y5, y6, y7, y8]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y3, y2, y4, y5, y4, 0, 0, y6, 3 y4]

  p' = s 4 - s 7   p = - s 4 + s 7


 
426 . Coloring, {3, 7, 11, 12}

R: [7, 7, 8, 6, A, A, A, C, B, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, y4, 2 y1, y1, y2, y7, y5, y6]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, y3, y4, y5, 2 y3 - 3 y7, 0, 0, y6, y7]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
427 . Coloring, {3, 8, 9, 10}

R: [7, 7, 8, 6, A, A, B, B, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y5, y4, y5, 0, y6, y7, y5]

  p' = - s 5 + s 8   p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y8, y7, 0, y6, y5, y6, y4, y3, y1, y2]

  p = s 3 - s 9


 
428 . Coloring, {3, 8, 9, 11}

R: [7, 7, 8, 6, A, A, B, B, C, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, 2 y4, y4, 0, y6, y7, y5]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[y5, y4, y3, 0, 0, y1, y2, y10, y9, y7, y8, y6]


 
429 . Coloring, {3, 8, 9, 12}

Ωp(Δ)=0:     p = s + s 2 + s 4 + 4s 5 - 8s 7 - 16s 8

R: [7, 7, 8, 6, A, A, B, B, C, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y3 + y2 + 2 y5 - y4, 0, 0, 0, 0, y2, y3, y2, y2 - y1 + y5, y1, y4, y5]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y3, y1, y2, y6, y7, y5, y4, 0, y9, y7, y8]

  p = - s 3 + s 10


 
430 . Coloring, {3, 8, 10, 11}

R: [7, 7, 8, 6, A, A, B, B, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y5, y3, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y6, 0, y1, 0, 0, y2, y5, y6, y3, y4, 0, y7]

  p = s 6 - s 8


 
431 . Coloring, {3, 8, 10, 12}

R: [7, 7, 8, 6, A, A, B, B, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y4, y3, y4, 2 y4, y5, y6, 0]

  p = s 4 - s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, 2 y4, y2, y4, y3, y4, 0, y5, 0, y6]

  p' = - s 2 + s 7   p = - s 2 + s 7


 
432 . Coloring, {3, 8, 11, 12}

R: [7, 7, 8, 6, A, A, B, B, B, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y7, y2, 0, y3, y4, y5, y6, 0, y8, 0, y9]


 
433 . Coloring, {3, 9, 10, 11}

R: [7, 7, 8, 6, A, A, B, C, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y2, 0, y3, -y2 + y3 - y4 + y1 - y7 + y5 - y6 + y8, y4, y1, y7, 0, y5, y6, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, 0, y6, y1, y2, y3, y9, y7, y8]


 
434 . Coloring, {3, 9, 10, 12}

R: [7, 7, 8, 6, A, A, B, C, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-5 y1 + 6 y3 - 5 y2 + 11 y4 - 5 y5 - 5 y6 + 11 y7, 5 y1, 0, 0, 0, 5 y3, 5 y2, 5 y3, 5 y4, 5 y5, 5 y6, 5 y7]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y3, y1, y2, y5, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9


 
435 . Coloring, {3, 9, 11, 12}

R: [7, 7, 8, 6, A, A, B, C, C, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, 0, y3, 2 y2, y2, y1, y5, y6, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y9, y8, 0]


 
436 . Coloring, {3, 10, 11, 12}

R: [7, 7, 8, 6, A, A, B, C, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y6, y7, y3, y4, y5, y8, y9]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y8, 0, y6, 0, y7, y3, y4, y5, 0, y1, y2, y9]


 
437 . Coloring, {4, 5, 6, 7}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, C, 1, 5]
B: [6, 8, 8, 6, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, 0, y4, 0, 0, y3, y1, y2]

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y1 + 8 y3 - 3 y4 + 5 y6, 0, -3 y2 + 5 y3 - 3 y5 + 8 y6, 0, 3 y1, 0, 3 y2, 3 y3, 3 y5, 3 y4, 3 y6]

  p = s - s 7   p' = s - s 7


 
438 . Coloring, {4, 5, 6, 8}

Ωp(Δ)=0:     p' = s 2 - 8s 5 - 16s 6   p' = s 3 - 8s 6 - 16s 7   p = s 2 - 8s 5 - 16s 6

R: [7, 7, 7, 7, 3, 3, B, B, B, C, 1, 5]
B: [6, 8, 8, 6, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 6 vs 6 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y6, 0, 0, 0, y4, y5]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y6, 0, y1, 0, y2, 0, y3, y4, y5, 0, y7]


 
439 . Coloring, {4, 5, 6, 9}

Ωp(Δ)=0:     p = s 2 + 2s 3 + 8s 5 + 32s 7   p' = s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, B, C, C, C, 1, 5]
B: [6, 8, 8, 6, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y3, 0, y1, 0, y2, 0, y4, 0, 0, 0, y5, y6]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y7, 0, y6, 0, y5, 0, y4, y3, y2, y1, 0]


 
440 . Coloring, {4, 5, 6, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, 2, 1, 5]
B: [6, 8, 8, 6, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y6, y2, 0, y3, 0, y4, 0, 0, 0, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 0, y3, y4, y5, y6, y7]


 
441 . Coloring, {4, 5, 6, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, C, 4, 5]
B: [6, 8, 8, 6, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, 0, 0, 0, y5, y6]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[2 y1, 11 y1 - 2 y2 + 11 y3 - 39 y4 + 11 y5 - 2 y6, 0, 0, 0, 2 y2, 0, 2 y3, 2 y4, 2 y5, 2 y6, 3 y1 + 3 y3 - 11 y4 + 3 y5 ]

  p = - s + s 7   p' = - s + s 7


 
442 . Coloring, {4, 5, 6, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, C, 1, 9]
B: [6, 8, 8, 6, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[2 y1, 0, 2 y2, 0, 0, 0, 2 y3, 0, 2 y4, 0, 2 y5, 3 y2]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y4, 0, y5, 0, y6, y7, y8]


 
443 . Coloring, {4, 5, 7, 8}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, C, 1, 5]
B: [6, 8, 8, 6, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, 0, 0, y6, y7, y5]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y3, 2 y1 - 2 y6, 0, y2, 0, y4, y5, y6, 2 y6, y7]

  p' = - s 6 + s 8   p = - s 6 + s 8


 
444 . Coloring, {4, 5, 7, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, C, 1, 5]
B: [6, 8, 8, 6, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y3, 0, y4, 0, y5, 0, y1, 0, 0, y2, 0, y6]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, 0, y5, 2 y6, y6, y7, 0]

  p = - s 3 + s 8


 
445 . Coloring, {4, 5, 7, 10}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, A, C, B, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y4, y5, y6, 0, y4, 0, y3, 0, 0, y1, y2, y2]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -3 y5 - 3 y1 - 3 y4 + 10 y3 - 3 y2, 3 y5, 0, 3 y1, 0, 3 y4, -3 y6 + 3 y3, 3 y6, 3 y2, 3 y3]

  p' = - s 2 + s 7   p = - s 2 + s 7


 
446 . Coloring, {4, 5, 7, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, A, C, B, C, 4, 5]
B: [6, 8, 8, 6, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y3, y1, 0, y7, 0, 0, y6, y4, y5]

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[3 y2, 3 y1, -3 y2 - 3 y1 - 3 y3 - 3 y8 + 13 y6 - 3 y7 - 3 y5 + 13 y4, 0, 0, 3 y3, 0, 3 y8, 3 y6, 3 y7, 3 y5, 3 y4]

  p = - s 3 - s 4 + s 8 + s 9


 
447 . Coloring, {4, 5, 7, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, A, C, B, C, 1, 9]
B: [6, 8, 8, 6, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, 0, y2 + y1 + y6 + y5 - y3 - y4, 0, y6, y5, y3, y4]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, 0, y6, 0, y7, y8, y9]


 
448 . Coloring, {4, 5, 8, 9}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, B, B, C, C, 1, 5]
B: [6, 8, 8, 6, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, 0, y4, 0, y7, y8, y9, y5, y6]


 
449 . Coloring, {4, 5, 8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, B, B, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y3, y1, 2 y1 - 3 y4, 0, 2 y4, 0, y2, 0, 0, y4, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 2 y2, 2 y1, 0, 2 y3, 0, 2 y4, 2 y5, 3 y1, 0, 2 y6]

  p = - s 5 + s 7


 
450 . Coloring, {4, 5, 8, 11}

Ωp(Δ)=0:     p = s 3 - 8s 6 - 16s 7   p' = s 3 - 8s 6 - 16s 7

R: [7, 7, 7, 7, 3, A, B, B, B, C, 4, 5]
B: [6, 8, 8, 6, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y4, y3, 0, y2, 0, 0, y7, y5, y6]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-4 y1 + 6 y3, 2 y1, 2 y2, 0, 0, 2 y3, 0, 2 y4, 2 y5, -6 y1 + 9 y3, 0, 2 y6]

  p = - s 5 + s 7   p' = s 5 - s 7


 
451 . Coloring, {4, 5, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, B, B, C, 1, 9]
B: [6, 8, 8, 6, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, 0, y2, 0, y3, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, 0, y6, 0, y7, 0, y8]


 
452 . Coloring, {4, 5, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y2 - 2 y5, 0, y4, 0, 0, y5, y6, 2 y5]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y3, y2, 0, y1, 0, y8, y7, y6, y5, y4]


 
453 . Coloring, {4, 5, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, C, 4, 5]
B: [6, 8, 8, 6, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[2 y1, 2 y7, 2 y2, 0, 0, 2 y3, 0, 2 y4, 2 y5, 3 y5, 2 y6, 0]

  p = - s 3 + s 8


 
454 . Coloring, {4, 5, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, C, 1, 9]
B: [6, 8, 8, 6, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y1, 0, 7 y1 + 7 y2 - 9 y3 + 7 y5 - 9 y4, 0, 0, 0, 2 y2, 0, 2 y3, 7 y1 + 7 y2 - 9 y3 + 7 y5 - 9 y4, 2 y5, 2 y4]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y5, y4, 0, y6, 0, y7, y8, 0]


 
455 . Coloring, {4, 5, 10, 11}

Ωp(Δ)=0:     p' = - 3s 2 + 2s 3 - 8s 5 + 32s 7   p = 3s 2 - 2s 3 + 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, B, C, B, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y3, y1, y2, 0, y4, 0, 0, y6, y5, y6]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[y7, 0, -y7 - y1 - y2 + y3 + y4 - y5 + y6, 0, 0, y1, 0, y2, y3, y4, y5, y6]

  p = - s 2 - s 3 + s 7 + s 8


 
456 . Coloring, {4, 5, 10, 12}

Ωp(Δ)=0:     p = 3s 2 + 10s 3 + 40s 5 + 32s 6 + 32s 7 + 128s 8   p' = - 3s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, B, C, B, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y5, y4, y3, 0, 0, 0, y2, 0, y4, y3, y1, y3]

  p' = - s 4 + s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -y1 + y2 - y3 - y4 + y5 - y6 + y7, y1, y2, y3, 0, y4, 0, y5, y6, y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
457 . Coloring, {4, 5, 11, 12}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7   p = 3s 2 - 10s 3 - 40s 5 + 32s 6 - 32s 7 + 128s 8

R: [7, 7, 7, 7, 3, A, B, C, B, C, 4, 9]
B: [6, 8, 8, 6, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, 0, y6, 0, y2, y3, y4, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y9, y8, y7, 0, y6, y5, 0, y4, 0, y3, y2, y1]


 
458 . Coloring, {4, 6, 7, 8}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, C, 1, 5]
B: [6, 8, 8, 6, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, y2, 0, y3, 0, 0, y5, 2 y4, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y6, y7, y8, 0, y2, 0, y1, y4, y5, 2 y7, y3]

  p = - s 7 + s 9


 
459 . Coloring, {4, 6, 7, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, C, 1, 5]
B: [6, 8, 8, 6, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, y1, 0, y2, 0, y3, 0, 0, y4, 0, y5]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y7, 0, y6, 2 y3, y4, y5, 0]

  p = - s 2 + s 8


 
460 . Coloring, {4, 6, 7, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, A, C, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2, y1, y5, 0, y2, 0, y3, 0, 0, y4, y5, y5]

  p' = - s 4 + s 7   p = s 3 - s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y2, y3, 0, y1, 0, y4, y5, y6, y7, y8]


 
461 . Coloring, {4, 6, 7, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, A, C, B, C, 4, 5]
B: [6, 8, 8, 6, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y5, y1, y2, 0, y3, 0, 0, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[11 y2 + 11 y1 + 11 y7 - 2 y6 - 39 y5 - 2 y4 + 11 y3, 2 y2, 2 y1, 0, 0, 2 y7, 0, 2 y6, 2 y5, 2 y4, 2 y3, 3 y2 + 3 y1 + 3 y7 - 11 y5 + 3 y3]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
462 . Coloring, {4, 6, 7, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, A, 3, A, C, B, C, 1, 9]
B: [6, 8, 8, 6, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, 0, y4, 0, y1, y2, y6 + y5 - y4 + y1 + y2 - y3, y3]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y9, y8, y7, y6, y5, 0, y4, 0, y3, y2, y1]


 
463 . Coloring, {4, 6, 8, 9}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, B, B, C, C, 1, 5]
B: [6, 8, 8, 6, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y1, 0, y2, 0, y3, 0, 4 y1 + 4 y2 + 5 y3 - 6 y4, 0, 0, -2 y1 - 2 y2 - 4 y3 + 5 y4, y4, 5 y1 + 5 y2 + 6 y3 - 8 y4]

  p' = s 2 - s 5   p = s 2 - s 5   p' = s 3 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y4, 0, y5, y6, y7, y8, y9]


 
464 . Coloring, {4, 6, 8, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, B, B, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y3, 0, y4, 0, 0, y6, y5, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, -2 y5 + 2 y4, 0, y4, y3, y2, 0, y1]

  p = - s 4 + s 6   p' = s 4 - s 6


 
465 . Coloring, {4, 6, 8, 11}

Ωp(Δ)=0:     p' = s 3 + 4s 4 - 8s 6 - 16s 7   p = s 3 - 16s 5 - 8s 6 + 16s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, B, B, B, C, 4, 5]
B: [6, 8, 8, 6, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 2 y1, -2 y1 + 9 y2 - 11 y3 - 2 y4, -2 y2 + 2 y3 + 2 y4, 0, 2 y2, 0, 0, 7 y2 - 9 y3 - 2 y4, 2 y4, 2 y3]

  p = - s 2 + s 5   p' = s 3 - s 6   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y1, y2, 0, 0, y3, 0, y4, y5, y6, 0, y7]

  p = s 6 - s 8


 
466 . Coloring, {4, 6, 8, 12}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 24s 6 - 16s 7 - 64s 8   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 7, 7, 7, A, 3, B, B, B, C, 1, 9]
B: [6, 8, 8, 6, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, 0, y2, 0, y3, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y2, y1, y4, y5, y3, 0, y6, 0, y7, 0, y8]


 
467 . Coloring, {4, 6, 9, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, y5, 0, y4, 0, y3, 0, 0, y7, y6, 2 y5]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y3, 0, y4, y5, y6, y7, y8]


 
468 . Coloring, {4, 6, 9, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, C, 4, 5]
B: [6, 8, 8, 6, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 16 y1, 16 y2, -35 y1 - 35 y2 + 28 y3 + 25 y4, 0, 16 y3, 0, 0, 16 y4, -25 y1 - 25 y2 + 20 y3 + 27 y4, 28 y1 + 28 y2 - 20 y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, 0, y5, 2 y3, y6, y7, 0]

  p = - s 2 + s 8


 
469 . Coloring, {4, 6, 9, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, C, 1, 9]
B: [6, 8, 8, 6, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y1, 0, 7 y1 + 7 y3 - 9 y2 + 7 y5 - 9 y4, 0, 0, 0, 2 y3, 0, 2 y2, 7 y1 + 7 y3 - 9 y2 + 7 y5 - 9 y4, 2 y5, 2 y4]

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y4, y5, y1, y2, 0, y6, 0, y7, y8, 0]


 
470 . Coloring, {4, 6, 10, 11}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, B, C, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y7, y2, y3, 0, y4, 0, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y4, y5, y6, y7, y8]


 
471 . Coloring, {4, 6, 10, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, B, C, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y2, y5, 0, 0, 0, y4, 0, y2, y5, y3, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y4, 0, y5, 0, y6, y7, y8]


 
472 . Coloring, {4, 6, 11, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, B, C, B, C, 4, 9]
B: [6, 8, 8, 6, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, 0, y2, 0, y4, y3, y6, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y4, y5, 0, y6, 0, y7, y8, y9]


 
473 . Coloring, {4, 7, 8, 9}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, C, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 8 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, 0, 0, y4, y5, y6]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, 0, y4, 0, y5, y6, 0, y7, y8]


 
474 . Coloring, {4, 7, 8, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, A, A, A, B, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2, y1, 0, 0, y5, 0, y4, 0, 0, y3, y5, 0]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y6, y3, 0, y4, 0, y5, y2, 0, y6 + y3 - y4 - y5 - y2 + y1, y1]

  p = - s 6 + s 7


 
475 . Coloring, {4, 7, 8, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, B, B, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, y5, 0, y4, 0, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, y2, -y1 + y6 + y5 + y4 - y3, 0, 0, y6, 0, y5, y4, 0, y2, y3]

  p = s 6 - s 8   p' = s 6 - s 7


 
476 . Coloring, {4, 7, 8, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, A, B, B, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 - y2 - y5 + y4 + y3, 0, 0, 0, 0, 0, y1, 0, y2, y5, y4, y3]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, y3, y2, y3 + y2 - y4, y3 + y2 - y1, y1, 0, y2, 0, 0, y3, y4]

  p' = s 6 - s 7   p = s 4 - s 8   p' = s 4 - s 7   p' = s 5 - s 7


 
477 . Coloring, {4, 7, 9, 10}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, A, A, A, C, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y5, y1, 0, 0, y4, 0, y3, 0, 0, y2, 0, y5]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y2, y1, 0, y3, 0, y6, y5, 0, y4, y7]


 
478 . Coloring, {4, 7, 9, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, C, C, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 5 6 vs 7

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, y5, y4, 0, y3, 0, 0, y2, 0, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y5, y2, 0, 0, y3, 0, y4, y5, 0, y6, 0]

  p = - s 2 + s 7


 
479 . Coloring, {4, 7, 9, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, A, A, A, C, C, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 5 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y2 - y1 - y3 + y4, 0, 0, 0, 0, 0, y2, 0, y1, y3, 0, y4]

  p = s 4 - s 5

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y3, y1, y2, y3, y4, 0, y5, 0, 0, y6, 0]

  p = - s 2 + s 7


 
480 . Coloring, {4, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, y4, y4, 0, y2, 0, 0, y3, y5, y5]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[5 y6, 0, 5 y5, 0, 0, 5 y4, 0, 5 y3, 5 y2, 0, -5 y6 - 5 y5 - 5 y4 - 5 y3 + 11 y2 + 11 y1, 5 y1]

  p = - s - s 2 + s 6 + s 7


 
481 . Coloring, {4, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 0, 0, y7, 0, y6, y5, y4, y3]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y6, y5, y4, 0, y3, 0, 0, y2, y7]


 
482 . Coloring, {4, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - y2 - y3 + y4 + y5, 0, 0, y1, 0, y2, y3, y4, y5]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 2 y3, y7, 0, y5, y6, 0, y4, 0, 0, y2, y3]

  p = - s 3 + s 8


 
483 . Coloring, {4, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y2, 0, y3, 0, y4, y5, y6, y7, y8]


 
484 . Coloring, {4, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 6 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y3, 10 y3 - 16 y1 + 4 y2, 0, 2 y2, 0, 0, 2 y1, 16 y3 - 27 y1 + 7 y2, 4 y3 + 3 y2 - 7 y1]

  p = - s + s 4   p' = - s + s 4   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, 0, y4, 0, y5, y6, y7, y8, y9]


 
485 . Coloring, {4, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-7 y4 + 9 y5 + 9 y1 - 7 y2 + 9 y3, 0, 0, 0, 0, 0, 7 y4, 0, 7 y5, 7 y1, 7 y2, 7 y3]

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 2 y2 - 2 y6, y1, y2, y3, y4, 0, y5, 0, 2 y6, y6, y7]

  p = - s 4 + s 8   p' = - s 4 + s 8


 
486 . Coloring, {4, 8, 10, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, B, B, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y3, 0, y2, y1, 0, y6, 0, 0, y5, y4, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y4, 0, y1 + y2 + y3 - y5, 0, 0, y1, 0, y2, y3, y4, 0, y5]

  p = - s 5 + s 6   p = - s 5 + s 7


 
487 . Coloring, {4, 8, 10, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, B, B, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, 0, y4, y4, y5, 0]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y1 + y2 + y3 - y5, y4, y1, y2, 0, y3, 0, y4, 0, y5]

  p = - s 3 + s 4 - s 5 + s 6   p = - s 3 + s 7


 
488 . Coloring, {4, 8, 11, 12}

Ωp(Δ)=0:     p = s 2 + 2s 3 - 8s 5 - 32s 7   p = - s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, A, A, B, B, B, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 5 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, 0, y3, y4, y5, y6]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y4, y1, y2 + y3 - y5, 0, y2, y1, 0, y3, 0, y4, 0, y5]

  p' = s 3 - s 7   p = s 3 - s 7   p' = s 4 - s 5 + s 6 - s 7


 
489 . Coloring, {4, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 2 + 10s 3 + 40s 5 + 32s 6 + 32s 7 + 128s 8   p' = - 3s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, B, C, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y3, y2, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, 0, y5, 0, y1, y2, y3, y4, y8]


 
490 . Coloring, {4, 9, 10, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 - 24s 5 - 16s 6 + 96s 7 + 64s 8   p = - 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, B, C, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 3 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y1 + 2 y3, y1, 0, 0, 0, 0, 2 y3, 0, y3, 2 y3 - y2, y2, y3]

  p = s 3 - s 5   p' = s 3 - s 4   p' = - s 4 + s 5   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y7, y1, y2, y6, 0, y3, 0, y4, y5, y8]


 
491 . Coloring, {4, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 2 + 2s 3 - 8s 5 + 32s 7   p' = 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, B, C, C, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 6 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, -y1 + 2 y4 - y3, 0, 0, y1, 0, -y2 + y4, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y6, y4, 0, y5, 0, y6, y7, 0]

  p = s 3 - s 8


 
492 . Coloring, {4, 10, 11, 12}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, C, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y4 - 2 y5, 0, y1, 0, 0, y2, 0, y4, 2 y5, y3, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, y7, y8]


 
493 . Coloring, {5, 6, 7, 8}

Ωp(Δ)=0:     p = - 9s 3 - 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, C, 1, 5]
B: [6, 8, 8, 7, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, 2 y4, y7]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 11 y3 - 5 y4 + 11 y5 - 5 y6 - 5 y7 - 5 y8 + 11 y1 - 5 y2, 0, 5 y3, 0, 5 y4, 5 y5, 5 y6, 5 y7, 5 y8, 5 y1, 5 y2]

  p = - s 5 - s 6 + s 8 + s 9


 
494 . Coloring, {5, 6, 7, 9}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, C, 1, 5]
B: [6, 8, 8, 7, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y3, 0, y1, 0, y2, y3, y4, 0, 0, y5, 0, y6]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y3, 0, y1, y4, y5, 2 y1, y7, y6, 0]

  p = - s 5 + s 8


 
495 . Coloring, {5, 6, 7, 10}

R: [7, 7, 7, 6, 3, 3, A, C, B, 2, 1, 5]
B: [6, 8, 8, 7, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 5 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, y6, y4, 0, 0, y5, y6, y6]

  p = s 4 - s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, 2 y5 - y1 - y4, 0, y2 + y3 - y5, y1, 2 y2 + 2 y3 - 2 y5, y2, y3, y4, y5]

  p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 6


 
496 . Coloring, {5, 6, 7, 11}

R: [7, 7, 7, 6, 3, 3, A, C, B, C, 4, 5]
B: [6, 8, 8, 7, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y5, y3, y4, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y2 - 3 y6 + 8 y3 - 3 y4 + 5 y7, -3 y1 + 5 y3 - 3 y5 + 8 y7, 0, 0, 0, 3 y1, 3 y2, 3 y6, 3 y3, 3 y4, 3 y5, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
497 . Coloring, {5, 6, 7, 12}

R: [7, 7, 7, 6, 3, 3, A, C, B, C, 1, 9]
B: [6, 8, 8, 7, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y2 + y3 - y4 - y7 + y5 + y6, 0, y1, 0, 0, y2, y3, 0, y4, y7, y5, y6]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y8, y4, y5, 0, y6, y7, y8]

  p = - s 6 + s 9


 
498 . Coloring, {5, 6, 8, 9}

R: [7, 7, 7, 6, 3, 3, B, B, C, C, 1, 5]
B: [6, 8, 8, 7, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, y3, y4, y2, 0, 0, 0, y1, 3 y4]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, y5, y6, y7, y9, y8]


 
499 . Coloring, {5, 6, 8, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, 2, 1, 5]
B: [6, 8, 8, 7, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y5, 2 y3, y4, 0, 2 y3, y3, y2, 0, 0, 0, y1, 0]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y1, 0, y1, y2, 2 y1, y3, y4, 0, y5]

  p' = s 4 - s 6   p = s 4 - s 6

 ยป SYNC'D !RANK'D


 
500 . Coloring, {5, 6, 8, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, C, 4, 5]
B: [6, 8, 8, 7, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y3, y1, 0, 0, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8


 
501 . Coloring, {5, 6, 8, 12}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, C, 1, 9]
B: [6, 8, 8, 7, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 0, y3, 0, 0, y2, y4, 0, 2 y3 - 2 y2, 0, y5, 2 y2]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, 2 y3, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 3 + s 8


 
502 . Coloring, {5, 6, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, 2, 1, 5]
B: [6, 8, 8, 7, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, 2 y1, y3, 0, y2, y1, y6, 0, 0, 0, y5, 2 y1]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, 2 y2, y4, y5, y6, y7]

  p = - s 2 + s 8


 
503 . Coloring, {5, 6, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, C, 4, 5]
B: [6, 8, 8, 7, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, 2 y4, y6, y7, 0]

  p = - s 2 + s 8


 
504 . Coloring, {5, 6, 9, 12}

Ωp(Δ)=0:     p = - 2s 2 - 5s 3 + 2s 4 + 16s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, C, 1, 9]
B: [6, 8, 8, 7, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 + 5 y4 - 3 y5 + 5 y6, 0, 3 y1, 0, 0, 3 y2, 3 y3, 0, 3 y4, 0, 3 y5, 3 y6]

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y1 + 3 y4 + y5 - y3 - y2 + y6, 2 y4, y4, y5, y3, 0, y2, y6, 0]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = - s 2 + s 8

 ยป SYNC'D !RANK'D


 
505 . Coloring, {5, 6, 10, 11}

R: [7, 7, 7, 6, 3, 3, B, C, B, 2, 4, 5]
B: [6, 8, 8, 7, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y7, y1, y2, y3, y5, y4, 0, 0, 0, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, 2 y3, y4, y5, y6, y7]

  p = - s 6 + s 8


 
506 . Coloring, {5, 6, 10, 12}

R: [7, 7, 7, 6, 3, 3, B, C, B, 2, 1, 9]
B: [6, 8, 8, 7, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 2 y4, y2, 0, 0, y4, y5, 0, y2, 0, y3, y4]

  p' = - s 4 + s 7   p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y3, 0, y5, y6, y7]

  p = - s 5 + s 8


 
507 . Coloring, {5, 6, 11, 12}

R: [7, 7, 7, 6, 3, 3, B, C, B, C, 4, 9]
B: [6, 8, 8, 7, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y6, y7, 0, y4, 0, y5, y3]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y4, y8, y5, 0, y6, y7, y8]

  p = - s 3 + s 9


 
508 . Coloring, {5, 7, 8, 9}

Ωp(Δ)=0:     p = - 2s 2 - 5s 3 + 2s 4 - 16s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, C, 1, 5]
B: [6, 8, 8, 7, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y3, 0, y1, y6, y4, 0, 0, y5, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y8, y7, y6, 0, y8 - y7, y5, y4, y3, y8 - y7, y2, y1 ]

  p' = - s 6 + s 9   p = - s 6 + s 9

 ยป SYNC'D !RANK'D


 
509 . Coloring, {5, 7, 8, 10}

R: [7, 7, 7, 6, 3, A, A, B, B, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2 + y3, y1, y2, 0, 2 y3, y3, y4, 0, 0, y5, 2 y3, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 11 y1 - 10 y5 + 11 y2 - 5 y3 - 5 y4 + 11 y6 - 5 y7, 5 y1, 0, 5 y5, 5 y2, 5 y3, 5 y4, 5 y5, 5 y6, 5 y7]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9


 
510 . Coloring, {5, 7, 8, 11}

R: [7, 7, 7, 6, 3, A, A, B, B, C, 4, 5]
B: [6, 8, 8, 7, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y3 + y4 - y7, y6, y1, 0, 0, y2, y5, y3, y4, y5, y6, y7]

  p' = s 8 - s 9   p' = s 7 - s 9   p = s 7 - s 10


 
511 . Coloring, {5, 7, 8, 12}

R: [7, 7, 7, 6, 3, A, A, B, B, C, 1, 9]
B: [6, 8, 8, 7, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2 - y3 - y4 + y5 + y6, 0, y1, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, -5 y1 + 11 y2 - 5 y9 - 5 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y7 - 5 y8, 5 y1, 5 y2, 5 y9, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = s 3 + s 4 + s 5 - s 8 - s 9 - s 10


 
512 . Coloring, {5, 7, 9, 10}

R: [7, 7, 7, 6, 3, A, A, C, C, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y3, y5, y1, 0, y2, y3, y4, 0, 0, y6, 0, 2 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -y3 + y5, y1, 0, y3, y2, 2 y3 - y5 + y6, y6, y3, y4, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
513 . Coloring, {5, 7, 9, 11}

R: [7, 7, 7, 6, 3, A, A, C, C, C, 4, 5]
B: [6, 8, 8, 7, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y3, y2, 0, 0, y4, y6, y5, 2 y6, y6, y7, 0]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
514 . Coloring, {5, 7, 9, 12}

R: [7, 7, 7, 6, 3, A, A, C, C, C, 1, 9]
B: [6, 8, 8, 7, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, y4, 0, 0, y4, y3, 0, y2, y1, 0, 2 y4 - y3 + y2 + y1]

  p' = s 4 - s 5   p = s 4 - s 6   p' = - s 5 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, y1, y2, 4 y1 - 2 y6, 2 y1 - y6, y3, y5, 0, y6, y7, 0]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
515 . Coloring, {5, 7, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y3, y2, y2, y3, y4, 0, 0, y5, y6, y6]

  p' = - s 5 + s 8   p = s 4 - s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[3 y1, 0, -3 y1 - 3 y2 - 3 y4 - 3 y3 + 10 y6 - 3 y5, 0, 0, 3 y2, 3 y4, 3 y3, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8


 
516 . Coloring, {5, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y3, y5, 0, 0, y5, y2, 0, y6, y7, y4, y5]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y4, y3, y2, -6 y4 - 6 y3 + 5 y2 + 4 y1 + 4 y6 - y5, y1, y6, 0, 5 y4 + 5 y3 - 4 y2 - 2 y1 - 2 y6, y5, -8 y4 - 8 y3 + 6 y2 + 5 y1 + 5 y6]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
517 . Coloring, {5, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, C, 4, 9]
B: [6, 8, 8, 7, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, 0, y5, y6, y7, y1 + y2 - y3 - y4 + y5 + y6 - y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y9, y8, y6, y7, 0, y4, y5, y6]

  p = s 5 - s 10


 
518 . Coloring, {5, 8, 9, 10}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 + 2 y6 - y4, y2, 0, y6 + y4, y6, y3, 0, 0, y4, y5, y6]

  p' = - s 5 + s 8   p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
519 . Coloring, {5, 8, 9, 11}

Ωp(Δ)=0:     p = 2s 2 - s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, C, 4, 5]
B: [6, 8, 8, 7, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y2 - y3 + y5 + y4 - y6 - y7, y1, y2, y3, y5, 0, 0, y4, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y9, y6, y10, y7, y8]


 
520 . Coloring, {5, 8, 9, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, C, 1, 9]
B: [6, 8, 8, 7, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 4 vs 8 8 vs 10

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-4 y3 + 3 y1 + 3 y2 + 3 y4, 0, 7 y3 - 3 y1 - 3 y2 - 3 y4, 0, 0, 7 y3 - 3 y1 - 3 y2 - 3 y4, 3 y3, 0, 3 y1, 3 y2, 3 y3, 3 y4]

  p = - s 3 + s 7   p' = - s 3 + s 7   p = - s 3 + s 5   p' = - s 3 + s 5

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y5, y4, y4 + y8, y3, y8, y1, y2, 0, y7, y8, y6]

  p = - s 4 + s 9   p' = - s 4 + s 9

 ยป SYNC'D !RANK'D


 
521 . Coloring, {5, 8, 10, 11}

R: [7, 7, 7, 6, 3, A, B, B, B, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y5 + y6 - y7, y1, y2, y3, y4, y5, 0, 0, y6, y7, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y4, 0, -8 y4 + y5 + y3 + 3 y2 - y1, 0, 0, -5 y4 + 2 y2, y4, y5, y3, y2, 0, y1]

  p' = - s 5 + s 7   p = s 5 - s 6   p' = - s 5 + s 6


 
522 . Coloring, {5, 8, 10, 12}

R: [7, 7, 7, 6, 3, A, B, B, B, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y5, y4, y3, 0, 0, y3, y2, 0, 2 y3, y1, y6, 0]

  p' = - s 4 + s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, 4 y1 - 2 y3, y2, 2 y1 - y3, y3, y4, 0, y5, 0, y6 ]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
523 . Coloring, {5, 8, 11, 12}

R: [7, 7, 7, 6, 3, A, B, B, B, C, 4, 9]
B: [6, 8, 8, 7, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y2 + y7 + y3 + y4 - y5 - y6, y1, 0, y2, y7, 0, y3, y4, y5, y6]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y3, y8, y7, 0, y6, y5, y3, y4, 0, y2, 0, y1]

  p = - s 4 + s 9


 
524 . Coloring, {5, 9, 10, 11}

R: [7, 7, 7, 6, 3, A, B, C, C, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, -y1 + y4 + y5 - y3 + y2 + y7 - y8 - y6, y4, y5, y3, y2, 0, 0, y7, y8, y6]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y9, y6, y7, y8]


 
525 . Coloring, {5, 9, 10, 12}

R: [7, 7, 7, 6, 3, A, B, C, C, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-y6 - 2 y1 - y2 + 6 y5 - y3 - y4, y6, y1, 0, 0, y1, y2, 0, y5, y3, y4, y5]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y4, y5, y1, y2, y3, y6, 0, y7, y8, y9]


 
526 . Coloring, {5, 9, 11, 12}

R: [7, 7, 7, 6, 3, A, B, C, C, C, 4, 9]
B: [6, 8, 8, 7, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, 0, y3, y2, y1, -y7 + y6 - y5 + y4 + y3 + y2 - y1]

  p = s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, y1, y5, 0, 2 y3, y4, y3, y7, 0, y6, y8, 0]

  p = - s 4 + s 9


 
527 . Coloring, {5, 10, 11, 12}

Ωp(Δ)=0:     p = - 9s 3 + 2s 4 - 16s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, B, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -2 y7 + y1 - y2 + y3 + y4 + y5 - y6, y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[-3 y1 + 7 y6 - 3 y2 - 3 y4 - 3 y5, 0, 3 y1, 0, -3 y3 + 3 y6, 3 y2, 3 y3, 3 y4, 0, 3 y6, 3 y5, 3 y6]

  p' = - s 3 + s 8   p' = - s 2 + s 7   p = - s 2 + s 7


 
528 . Coloring, {6, 7, 8, 9}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, C, 1, 5]
B: [6, 8, 8, 7, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y3, 0, y3 - y5, 0, y2, y5, y1, 0, 0, y6, y5, y4]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y8, y9, 0, y8, y7, y6, y5, y4, y3, y2]

  p = s 7 - s 10


 
529 . Coloring, {6, 7, 8, 10}

R: [7, 7, 7, 6, A, 3, A, B, B, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[2 y4, y1, y4, 0, 2 y3, y3, y2, 0, 0, y5, 2 y3, 0]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 5 y6, 5 y5, 0, 5 y6, 5 y4, 5 y3, 5 y2, -5 y6 + 5 y3, 5 y1, -5 y6 + 11 y5 + 11 y4 - 10 y3 - 5 y2 + 11 y1]

  p' = s 4 + s 5 - s 7 - s 8   p' = s 3 - s 5 - s 6 + s 8   p = s 3 - s 9


 
530 . Coloring, {6, 7, 8, 11}

R: [7, 7, 7, 6, A, 3, A, B, B, C, 4, 5]
B: [6, 8, 8, 7, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y2, y3, y1, y7, y8, 0, 0, y6, y4, y5]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 + y2 - y4 - y5 - y6 + y7 + y8, y1, y3, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 8 + s 9   p = - s 8 + s 10


 
531 . Coloring, {6, 7, 8, 12}

R: [7, 7, 7, 6, A, 3, A, B, B, C, 1, 9]
B: [6, 8, 8, 7, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y4 + y1 + y2 - y3 - y5 + y6 + y7, 0, y4, 0, 0, y1, y2, 0, y3, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 5 y7, 5 y5, 5 y6, 5 y4, 5 y2, 5 y3, 5 y1, 0, -5 y7 - 5 y5 + 11 y6 - 5 y4 - 5 y2 + 11 y3 - 5 y1 + 11 y9 - 5 y8, 5 y9, 5 y8]

  p = - s 4 - s 5 - s 6 + s 8 + s 9 + s 10


 
532 . Coloring, {6, 7, 9, 10}

R: [7, 7, 7, 6, A, 3, A, C, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[2 y3, y1, y2, 0, 2 y2, y3, y4, 0, 0, y5, 0, 2 y3]

  p = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3 - y5, y1, 0, y3 - y5, y2, y3, y4, y5, y6, y7]

  p' = s 5 - s 8   p = - s 5 + s 8


 
533 . Coloring, {6, 7, 9, 11}

R: [7, 7, 7, 6, A, 3, A, C, C, C, 4, 5]
B: [6, 8, 8, 7, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 - 2 y3 + y2 - y4 - y5 + y6, y1, y3, 0, 0, y2, y3, y4, 2 y3, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8   p' = - s 2 + s 8


 
534 . Coloring, {6, 7, 9, 12}

R: [7, 7, 7, 6, A, 3, A, C, C, C, 1, 9]
B: [6, 8, 8, 7, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y1, 0, -y1 + y2 - y3 - y4 + y5, 0, 0, y1, y2, 0, y3, y4, 0, y5]

  p = - s 5 + s 6   p = - s 5 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y3, y4, y2, -2 y4 + 4 y6, -y4 + 2 y6, y1, y7, 0, y6, y5, 0]

  p' = - s 5 + s 8   p = s 5 - s 8


 
535 . Coloring, {6, 7, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y7, y7, y3, y4, 0, 0, y5, y6, y6]

  p' = - s 5 + s 8   p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 - y3 - y4 - y5 + y6 + y7, 0, y2, 0, 0, y1, y2, y3, y4, y5, y6, y7]

  p = - s 7 + s 8   p = - s 7 + s 9

 ยป SYNC'D !RANK'D


 
536 . Coloring, {6, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, -y6 + y7, 0, 0, y6, y3, 0, y7, y4, y5, y6]

  p = s 5 - s 8   p' = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
537 . Coloring, {6, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, C, 4, 9]
B: [6, 8, 8, 7, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 + y4 - y5 - y6 + y7, y1, y2, 0, y3, y4, y8, y5, 0, y6, y7, y8]

  p = - s 4 + s 10   p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
538 . Coloring, {6, 8, 9, 10}

Ωp(Δ)=0:     p = 2s 2 + s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3 - y7, 0, y3, y7, y4, 0, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y2, y3, y6, y4, y5, y8, y7]

  p = - s 3 + s 9


 
539 . Coloring, {6, 8, 9, 11}

Ωp(Δ)=0:     p = - 2s 2 + s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, C, 4, 5]
B: [6, 8, 8, 7, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -3 y1 + 5 y2 - 3 y3 - 3 y4 + 5 y5 - 3 y6 + 5 y7, 3 y1, 3 y2, 3 y3, 3 y4, 0, 0, 3 y5, 3 y6, 3 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y4, 0, 0, y3, y4, y5, y6, y7, y8, y9]

  p = - s 2 + s 10


 
540 . Coloring, {6, 8, 9, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, C, 1, 9]
B: [6, 8, 8, 7, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-3 y1 - 3 y3 - 3 y2 + 10 y5 - 3 y4, 0, 3 y1, 0, 0, 3 y3, 3 y2, 0, -3 y3 + 3 y5, 3 y3, 3 y4, 3 y5]

  p' = - s 4 + s 7   p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, y4, y8, y5, y6, 0, y9, y8, y7]

  p = s 6 - s 10


 
541 . Coloring, {6, 8, 10, 11}

R: [7, 7, 7, 6, A, 3, B, B, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 5 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y8, y7, y5, y6, y4, y3, 0, 0, y1, y2, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, y1, 0, 0, -3 y1 + 2 y5, y1, y5, y4, y3, 0, y2]

  p' = - s 5 + s 7   p' = s 4 - s 6   p = s 4 - s 6


 
542 . Coloring, {6, 8, 10, 12}

R: [7, 7, 7, 6, A, 3, B, B, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y2 + y4, y2, 0, 0, y4, y3, 0, 2 y4, y4, y5, 0]

  p' = s 3 - s 6   p' = s 4 - s 7   p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y2, 2 y3, y1, y3, y4, y7, 0, y6, 0, y5]

  p = - s 4 + s 8


 
543 . Coloring, {6, 8, 11, 12}

R: [7, 7, 7, 6, A, 3, B, B, B, C, 4, 9]
B: [6, 8, 8, 7, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y4, y5, 0, y6, y7, y8, y3]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[2 y2, y1, y5, 0, y4, y3, y2, y7, 0, y6, 0, y8]

  p = - s 5 + s 9


 
544 . Coloring, {6, 9, 10, 11}

R: [7, 7, 7, 6, A, 3, B, C, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, 0, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y2, y3, y4, y7, y5, y6, y8]

  p = - s 3 + s 9


 
545 . Coloring, {6, 9, 10, 12}

R: [7, 7, 7, 6, A, 3, B, C, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-2 y1 - 3 y3 - y2 + 6 y5 - y4, y1 + y3, y1, 0, 0, y3, y2, 0, y5, y3, y4, y5]

  p' = - s 4 + s 7   p = - s 3 + s 9   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y9, y8, y7, y6, y5, y4, 0, y3, y2, y1]


 
546 . Coloring, {6, 9, 11, 12}

R: [7, 7, 7, 6, A, 3, B, C, C, C, 4, 9]
B: [6, 8, 8, 7, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, 0, 7 y7, 7 y6, 0, 7 y5, 7 y4, 0, 7 y3, 7 y2, -7 y7 - 7 y6 - 7 y5 - 7 y4 + 9 y3 + 9 y2 + 9 y1, 7 y1]

  p = s 2 + s 3 - s 7 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y5, y4, y5, y6, 0, y7, y8, 0]

  p = - s 3 + s 9


 
547 . Coloring, {6, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, C, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y5, y1, y2, 0, y3, y4, 0, y5, y7, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
548 . Coloring, {7, 8, 9, 10}

R: [7, 7, 7, 6, A, A, A, B, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2, y1, 0, 0, y2, y5, y3, 0, 0, y4, y5, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y7, 0, y6, y8]


 
549 . Coloring, {7, 8, 9, 11}

R: [7, 7, 7, 6, A, A, A, B, C, C, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, y1, y2, 3 y4, 0, 0, y3, y4, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 2 y3, y2, 0, 0, y4, y3, y7, y8, 0, y6, y5]

  p = - s 2 + s 9


 
550 . Coloring, {7, 8, 9, 12}

R: [7, 7, 7, 6, A, A, A, B, C, C, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 6 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y4 + y1 - y2 - y3 + y5, 0, 0, 0, 0, y4, y1, 0, y2, y3, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 10 y3, 5 y4, 5 y2, 5 y1, 5 y3, 6 y4 - 5 y2 + 12 y3 + 6 y6 - 5 y5, 5 y4 - 5 y1 + 5 y3 + 5 y6, 0, 0, 5 y5, 5 y6]

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7   p = s 2 - s 5 - s 6 + s 9


 
551 . Coloring, {7, 8, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, 0, y4, y6, y2, y3, 0, 0, y1, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, 0, y7, y8]


 
552 . Coloring, {7, 8, 10, 12}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y5, 0, 0, 0, y3, y4, 0, 2 y3, y2, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 5 y2, 5 y1, -5 y2 + 11 y1 - 5 y3 + 11 y4 - 5 y5 + 11 y6 - 5 y7, 5 y3, 5 y4, 5 y5, 0, 0, 5 y6, 5 y7]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

 ยป SYNC'D !RANK'D


 
553 . Coloring, {7, 8, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 + y5 - y3 - y4 + y2 + y6, 0, y1, y5, 0, y3, y4, y2, y6]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y2, 2 y3, y1, 0, y5, y4, y3, y6, 0, 0, y8, y7]

  p = - s 5 + s 9


 
554 . Coloring, {7, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, 2 y2 - 2 y3, y2, y3, y4, 0, 0, y5, 0, 2 y2 - 2 y3]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y7, y6, 0, y5, 2 y4]

  p = s 3 - s 8


 
555 . Coloring, {7, 9, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 2 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[2 y1, -2 y1 + 2 y2, 0, 0, 0, y1, -y1 + 2 y2, 0, y2, 2 y2, 0, y2]

  p = - s 2 + s 4   p = - s 2 + s 7   p = - s 2 + s 5   p = - s 2 + s 6   p = - s 2 + s 3

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y6, y5, y3, y4, y2, y1, 0, 0, y7, 2 y4]

  p = - s 5 + s 8


 
556 . Coloring, {7, 9, 11, 12}

Ωp(Δ)=0:     p = - 2s 2 + s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 4 vs 6 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, -2 y1 + 2 y2 + 2 y3 - 2 y4, 0, y1, -3 y1 + 3 y2 + 3 y3 - 3 y4, 0, y2, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, 2 y4, y1, 0, 2 y4, y6, y4, y5, 0, 0, y3, 0]

  p = - s 2 + s 7   p' = - s 2 + s 7

 ยป SYNC'D !RANK'D


 
557 . Coloring, {7, 10, 11, 12}

R: [7, 7, 7, 6, A, A, A, C, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y5, 0, y4, y6, y8, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y3, y4, y5, y6, 0, 0, y7, 3 y5]

  p = - s 3 + s 8


 
558 . Coloring, {8, 9, 10, 11}

R: [7, 7, 7, 6, A, A, B, B, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y7, 0, y6, y5, y4, y3, 0, 0, y2, y1, y7 - y6 + y5 + y4 - y3 - y2 + y1]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y9, y8]


 
559 . Coloring, {8, 9, 10, 12}

R: [7, 7, 7, 6, A, A, B, B, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y7 - 3 y6 - 3 y5 + 13 y4 - 3 y3 - 3 y2 + 13 y1, 3 y7, 0, 0, 0, 3 y6, 3 y5, 0, 3 y4, 3 y3, 3 y2, 3 y1]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y6, y4, y5, 0, y8, y6, y7]

  p = s 5 - s 9


 
560 . Coloring, {8, 9, 11, 12}

R: [7, 7, 7, 6, A, A, B, B, C, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 6 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, 0, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y5, y3 + y6, y2 + y1 - y6 - y4, 0, y2, y3, y6, y1, 0, y5, y6, y4]

  p = s 4 - s 5 - s 7 + s 10   p = s 4 - s 5 + s 6 - s 7   p = - s 4 + s 8   p' = - s 4 + s 8


 
561 . Coloring, {8, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, B, B, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, 0, y5, y6, y7, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y1, 0, y2, 0, y3, -3 y1 + 2 y5, y1, y4, 0, y5, 0, y6]

  p = s 3 - s 7   p' = s 3 - s 7


 
562 . Coloring, {9, 10, 11, 12}

Ωp(Δ)=0:     p = 6s 2 + s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, A, B, C, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + 2 y4, 0, y1, 0, -y2 + 2 y4, y2, 0, y4, 2 y4 - y3, y3, y4]

  p' = - s 2 + s 4 - s 5 + s 7   p' = - s 2 + s 3 - s 5 + s 6   p' = s - s 2 + s 4 - s 5   p = s - s 3 + s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
563 . Coloring, {2, 3, 4, 5, 6}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, B, C, B, C, 1, 5]
B: [6, 7, 7, 6, A, A, A, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[7 y1, 0, 9 y1 - 7 y2 + 9 y3 - 7 y4 + 9 y5 - 7 y6, 0, 7 y2, 0, 7 y3, 7 y4, 0, 0, 7 y5, 7 y6]

  p = s + s 2 + s 3 - s 5 - s 6 - s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 0, 3 y1, 0, 3 y2, 3 y3, 0, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8


 
564 . Coloring, {2, 3, 4, 5, 7}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, A, A, C, B, C, 1, 5]
B: [6, 7, 7, 6, A, 3, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -3 y4 - 3 y5 - 3 y2 - 3 y3 + 13 y1 - 3 y6 - 3 y7 + 13 y8, 3 y4, 3 y5, 0, 3 y2, 3 y3, 0, 3 y1, 3 y6, 3 y7, 3 y8]

  p = - s 3 - s 4 + s 8 + s 9


 
565 . Coloring, {2, 3, 4, 5, 8}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, B, B, C, 1, 5]
B: [6, 7, 7, 6, A, 3, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y8, y6, y7]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -y1 - y2 - y3 - y4 + 6 y6 - y5, y1, y2, 0, y3, y4, 0, y6, y5, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
566 . Coloring, {2, 3, 4, 5, 9}

Ωp(Δ)=0:     p = - s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, C, C, C, 1, 5]
B: [6, 7, 7, 6, A, 3, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 8 8 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[2 y3, 0, 5 y3 - 2 y4, 0, 5 y3 - 2 y1 - 2 y2, 0, 2 y3, 2 y1, 0, 2 y2, 2 y3, 2 y4]

  p = - s 2 + s 3 - s 4 + s 5   p = - s 2 + s 6   p' = - s 2 + s 6   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y8, y2, 0, y3, y4, 0, y5, y6, y7, 0]


 
567 . Coloring, {2, 3, 4, 5, 10}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 8s 6 - 16s 7 + 64s 8   p = 9s 2 + 2s 4 - 16s 6 + 16s 7 - 96s 8   p = - 27s 2 + 8s 5 + 48s 6 - 64s 7 + 256s 8

R: [7, 8, 8, 7, 3, A, B, C, B, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[7 y1, 9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y6 + 9 y7 - 7 y8, 7 y2, 0, 7 y3, 0, 7 y4, 7 y5, 0, 7 y6, 7 y7, 7 y8]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y2, -y2 + y1 + y3 - y4 - y5 + y6 + y7, 0, y1, y3, 0, y4, y5, y6, y7]

  p = - s 7 + s 8


 
568 . Coloring, {2, 3, 4, 5, 11}

Ωp(Δ)=0:     p = 3s 2 + 14s 3 - 24s 5 + 32s 6 - 32s 7 - 128s 8   p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, B, C, B, C, 4, 5]
B: [6, 7, 7, 6, A, 3, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 0, 7 y7, 7 y6, 7 y5, 0, 7 y4, 7 y3, 0, 7 y2, 7 y1, -7 y7 + 9 y6 - 7 y5 + 9 y4 - 7 y3 - 7 y2 + 9 y1]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y8 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 3 y1, 3 y2, 0, 0, 3 y3, 3 y8, 0, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 5 - s 6 + s 8 + s 9


 
569 . Coloring, {2, 3, 4, 5, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 - 24s 5 + 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, A, B, C, B, C, 1, 9]
B: [6, 7, 7, 6, A, 3, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, y4, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
570 . Coloring, {2, 3, 4, 6, 7}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7   p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, A, 3, A, C, B, C, 1, 5]
B: [6, 7, 7, 6, 3, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y3, 0, y4, 0, y1, 0, y2, y3, 0, y6, y4, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -2 y1 + 11 y5 - 2 y4 + 11 y2 + 11 y3 - 2 y6 - 39 y7, 2 y1, 2 y5, 0, 2 y4, 2 y2, 0, 3 y5 + 3 y2 + 3 y3 - 11 y7, 2 y3, 2 y6, 2 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
571 . Coloring, {2, 3, 4, 6, 8}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, B, B, B, C, 1, 5]
B: [6, 7, 7, 6, 3, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-2 y1 + 2 y3 + 2 y2 + 2 y5 - 2 y4, 0, 2 y1, 0, 5 y3 + 5 y2 - 2 y5 - 7 y4, 0, 2 y3, 2 y2, 0, 2 y5, 7 y3 + 7 y2 - 2 y5 - 9 y4, 2 y4]

  p = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y3, y2, 2 y2, 0, y1, -y3 - 3 y2 - y1 - y4 + 6 y5, 0, y5, y4, 0, y5]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6


 
572 . Coloring, {2, 3, 4, 6, 9}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, B, C, C, C, 1, 5]
B: [6, 7, 7, 6, 3, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y5, 0, y4, 0, y3, 0, y5, y2, 0, y1, y5, 5 y5 - y4 - y3 - y2 - y1]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, 0, 2 y2, y7, y6, 0]

  p = - s 5 + s 8


 
573 . Coloring, {2, 3, 4, 6, 10}

Ωp(Δ)=0:     p = s 2 + 8s 5 + 16s 6   p' = s 3 + 8s 6 + 16s 7   p' = s 2 + 8s 5 + 16s 6

R: [7, 8, 8, 7, A, 3, B, C, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[7 y1, 9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y8 + 9 y6 - 7 y7, 7 y2, 0, 7 y3, 0, 7 y4, 7 y5, 0, 7 y8, 7 y6, 7 y7]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y5, y4, 0, y3, y4, 0, y2, y1, y5, -y3 + y2 + y1]

  p' = - s 6 + s 7   p' = s 5 - s 6   p = s 5 - s 7


 
574 . Coloring, {2, 3, 4, 6, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, B, C, B, C, 4, 5]
B: [6, 7, 7, 6, 3, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 3 y1, 7 y5 - 3 y2 - 3 y4, -3 y1 + 3 y5, 0, 3 y2, 3 y5 - 3 y3, 0, 3 y3, 3 y4, 3 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 6 y6 - 3 y2 - 3 y3 + 13 y4 - 3 y5 + 13 y7, 3 y1, 3 y6, 0, 0, 3 y2, 3 y3, 0, 3 y4, 3 y5, 3 y6, 3 y7]

  p = s 4 + s 5 - s 7 - s 8   p' = - s 4 - s 5 + s 7 + s 8


 
575 . Coloring, {2, 3, 4, 6, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, B, C, B, C, 1, 9]
B: [6, 7, 7, 6, 3, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, y5, y4, y4, y3, y2, 0, 0, y1, -y5 + y3, -y5 + y3]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
576 . Coloring, {2, 3, 4, 7, 8}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, A, A, B, B, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, y4, 0, y6, y5, y7]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, y2, y3, y4, 0, y5, y6, 0, y1, 0, -y2 - y3 - y4 - y5 - y6 + 6 y1, y1]

  p' = s 2 - s 7   p = s 2 - s 7


 
577 . Coloring, {2, 3, 4, 7, 9}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, C, C, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[y5, 0, 0, 0, y4, 0, y3, y5, 0, y2, 0, y1]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y1, y2, 0, y4, y5, 0, y6, 0, y3, 0]

  p = - s 2 + s 7


 
578 . Coloring, {2, 3, 4, 7, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y8, y1, 0, 0, y2, 0, y3, y4, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, 0, 5 y1, -5 y1 - 5 y6 - 5 y5 + 11 y4 - 5 y3 + 11 y2, 0, 5 y6, 5 y5, 0, 5 y4, 0, 5 y3, 5 y2]

  p = - s - s 2 + s 6 + s 7


 
579 . Coloring, {2, 3, 4, 7, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, y5, 0, y4, 2 y2, 0, y3, y2, y1]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y5, 3 y4, 3 y3, 0, 0, 3 y2, -3 y5 - 3 y4 - 3 y3 - 3 y2 + 13 y1 - 3 y7 + 13 y6, 0, 3 y1, 0, 3 y7, 3 y6]

  p = - s 2 - s 3 + s 7 + s 8


 
580 . Coloring, {2, 3, 4, 7, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y3, 0, 0, 0, 0, 0, y1, y2, y7, y6, y4, y5]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y1, y7, y6, y5, y4, y3, 0, 0, 0, y2, y1]

  p = - s 3 + s 8


 
581 . Coloring, {2, 3, 4, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, B, B, C, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[2 y1, 0, 0, 0, 2 y2, 0, 7 y1 - 27 y2 - 2 y3 + 16 y4, 2 y3, 0, 3 y1 - 7 y2 + 4 y4, 2 y4, 4 y1 - 16 y2 + 10 y4]

  p' = - s 3 + s 6   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8, y9]


 
582 . Coloring, {2, 3, 4, 8, 10}

Ωp(Δ)=0:     p = s 2 - 8s 5 - 16s 6   p' = s 2 - 8s 5 - 16s 6   p' = s 3 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, A, B, B, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 7 vs 7 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, y5, 0, y4, y3, 0, y7, y6, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 + y3 - y5 - y6 + y4, y1, 0, y2, y3, 0, y5, y6, 0, y4]

  p = s 6 - s 7


 
583 . Coloring, {2, 3, 4, 8, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, A, A, B, B, B, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 4 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, -y1 - y2 - y3 + 5 y4, y4, 0, y1, y2, 0, y4, y3, y4]

  p' = - s 2 + s 5   p = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-y2 + 2 y4, -y1 + 2 y4, y1, 0, 0, 2 y4 - y3, y2, 0, y4, y3, 0, y4]

  p = s 4 - s 8   p' = s 6 - s 7   p' = s 5 - s 7   p' = s 4 - s 7


 
584 . Coloring, {2, 3, 4, 8, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, B, B, B, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y4, y3, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y6, y3, y3, y4, 0, 0, y5, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
585 . Coloring, {2, 3, 4, 9, 10}

Ωp(Δ)=0:     p = s 2   p' = s 2   p' = s 3   p' = s 4   p' = s 5   p' = s 6   p' = s 7

R: [7, 8, 8, 7, A, A, B, C, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 8 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[y1, y1, 0, 0, y1, 0, y1, y1, 0, y1, y1, y1]

  p = - s + s 5   p = - s + s 4   p = - s + s 7   p = - s + s 6   p = - s + s 8   p = - s + s 3   p = - s + s 2

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1, 0, y1, y1, 0, y1, y1, y1, y1]

  p = s - s 2   p' = - s + s 3   p' = - s + s 2   p' = - s + s 4   p' = - s + s 5   p' = - s + s 7   p' = - s + s 6


` See 8-level graph

`

M              \ ;       N

$ [ [0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1] , [1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1] , [0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1] , [0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1] , [1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1] , [0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2] , [1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1] , [0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0] ] $     $ [ [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0] ] $

τ= 20 , r'= 7/8

R: [7, 8, 8, 7, A, A, B, C, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, B, B, C, 4, 9]

Ranges

Action of R on ranges, [[1], [1]]
Action of B on ranges, [[2], [2]]

Cycles:    R , {{2, 5, 8, 10, 12}, {1, 7, 11}},   B , {{3, 4, 6, 7, 9, 10, 11, 12}}

β({1, 2, 5, 7, 8, 10, 11, 12}) = 1/2
β({3, 4, 6, 7, 9, 10, 11, 12}) = 1/2

Partitions
α([{7}, {10}, {11}, {2, 3}, {12}, {8, 9}, {5, 6}, {1, 4}]) = 1/1

b1 = {7} ` , ` b2 = {10} ` , ` b3 = {11} ` , ` b4 = {2, 3} ` , ` b5 = {12} ` , ` b6 = {8, 9} ` , ` b7 = {5, 6} ` , ` b8 = {1, 4}

Action of R and B on the blocks of the partitions: = [8, 7, 1, 2, 6, 4, 5, 3] [4, 1, 6, 7, 2, 5, 8, 3]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-8 partition graph.

`

Right Group
Coloring {2, 3, 4, 9, 10}
Rank8
R,B [7, 8, 8, 7, A, A, B, C, C, 2, 1, 5], [6, 7, 7, 6, 3, 3, A, B, B, C, 4, 9]
π2 [1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2]
u2 [1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 1)
wpp [2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1]
π8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

 

 
586 . Coloring, {2, 3, 4, 9, 11}

Ωp(Δ)=0:     p = 3s 2 + 2s 3 + 8s 5 + 32s 7   p = - 9s 2 + 4s 4 - 24s 5 + 16s 6 - 96s 7 + 64s 8

R: [7, 8, 8, 7, A, A, B, C, C, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 8 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y3, -y1 - y2 + 5 y3 - y4, 0, y3, y1, 0, y2, y3, y4]

  p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, 0, y6, y7, y8, 0]


 
587 . Coloring, {2, 3, 4, 9, 12}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7   p = 9s 2 - 28s 4 + 40s 5 - 16s 6 - 96s 7 + 64s 8

R: [7, 8, 8, 7, A, A, B, C, C, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 3 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, 0, 0, y2, y1, 5 y2 - 2 y1 - y3, y1, y2, y3]

  p' = - s 2 + s 4   p = - s 2 + s 6   p' = - s 2 + s 6   p = - s 2 + s 4

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y5, y3, y7, y4, y2, 0, 0, y6, y7, 0]

  p = - s 5 + s 8


 
588 . Coloring, {2, 3, 4, 10, 11}

Ωp(Δ)=0:     p = s 3 + 8s 6 + 16s 7   p' = s 3 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, A, B, C, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, 9 y1 - 7 y2 + 9 y3 - 7 y4 - 7 y5 + 9 y6 - 7 y7, 0, 7 y1, 7 y2, 0, 7 y3, 7 y4, 0, 7 y5, 7 y6, 7 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[-y2 + y1 + y5 - y3 - y4 + y7 + y6, 0, y2, 0, 0, y1, y5, 0, y3, y4, y7, y6]

  p = - s 7 + s 8


 
589 . Coloring, {2, 3, 4, 10, 12}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 8s 6 - 16s 7   p' = s 3 - 8s 5 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, A, B, C, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y8, 0, 0, 0, 0, y7, y6, y5, y4, y3, y2]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, 0, 0, y3, y2, y1]


 
590 . Coloring, {2, 3, 4, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, B, C, B, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y4, 0, 0, y1, y3, y2, y3, y5, y6]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y2, y1, 0, y3, y7, y6, 0, 0, y5, y4, y4]

  p' = s 5 - s 8   p = s 5 - s 8


 
591 . Coloring, {2, 3, 5, 6, 7}

R: [7, 8, 8, 6, 3, 3, A, C, B, C, 1, 5]
B: [6, 7, 7, 7, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y3, 0, y2, 0, y1, y6, y7, y8, 0, y5, y6, y4]

  p = s 5 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y5 + 13 y3 - 3 y4 - 3 y6 + 13 y7, 0, 3 y1, 0, 3 y2, 3 y5, 0, 3 y3, 3 y4, 3 y6, 3 y7]

  p = s 4 + s 5 - s 7 - s 8


 
592 . Coloring, {2, 3, 5, 6, 8}

Ωp(Δ)=0:     p = 2s 2 - s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, C, 1, 5]
B: [6, 7, 7, 7, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 2 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y4, y6, y2, y3, 0, 0, y7, 2 y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 2 y2 - 2 y1, 0, -2 y2 + 4 y1, 0, -y2 + 2 y1, y2, 0, y1, 2 y1, 0, y1]

  p' = s 4 - s 6   p' = s 3 - s 6   p' = s 2 - s 6   p = s 2 - s 7   p' = s 5 - s 6


 
593 . Coloring, {2, 3, 5, 6, 9}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, C, 1, 5]
B: [6, 7, 7, 7, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[5 y1, 0, 11 y1 - 5 y2 - 5 y3 + 11 y7 - 5 y6 + 11 y4 - 5 y5, 0, 5 y2, 5 y3, 5 y7, 5 y6, 0, 0, 5 y4, 5 y5]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, 0, y4, 0, y5, y6, 0, 2 y5, y1, y2, 0]

  p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
594 . Coloring, {2, 3, 5, 6, 10}

R: [7, 8, 8, 6, 3, 3, B, C, B, 2, 1, 5]
B: [6, 7, 7, 7, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 5 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[5 y3, 10 y6, 5 y2, 0, 5 y1, 5 y6, -5 y3 + 12 y6 + 6 y2 + 6 y5 - 5 y4, 5 y2 - 5 y1 + 5 y6 + 5 y5, 0, 0, 5 y4, 5 y5]

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7   p = s 2 - s 5 - s 6 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, 0, y4, y3, 0, y2, y1, y4, y5 - y3 + y2 + y1 - 2 y4]

  p' = s 5 - s 6   p = s 5 - s 7


 
595 . Coloring, {2, 3, 5, 6, 11}

R: [7, 8, 8, 6, 3, 3, B, C, B, C, 4, 5]
B: [6, 7, 7, 7, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y2, y1, y2 + y1 - y5 + y4 - y3 - y7 + y6, y5, y4, y3, 0, 0, y7, y6]

  p = s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 3 y2 - 3 y3 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 0, 0, 0, 3 y2, 3 y3, 0, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8


 
596 . Coloring, {2, 3, 5, 6, 12}

R: [7, 8, 8, 6, 3, 3, B, C, B, C, 1, 9]
B: [6, 7, 7, 7, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, y4, y2, y3, y1, 0, y7, y8]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y3, 0, y2, y2, y5, y1, 0, 0, y4, y5, y5]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6


 
597 . Coloring, {2, 3, 5, 7, 8}

R: [7, 8, 8, 6, 3, A, A, B, B, C, 1, 5]
B: [6, 7, 7, 7, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, y1 + y3, y1, y2, 0, y3, -2 y1 - 3 y3 - y2 - y4 + 6 y5, 0, y5, y3, y4, y5]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = s 3 - s 6   p' = s 3 - s 6


 
598 . Coloring, {2, 3, 5, 7, 9}

R: [7, 8, 8, 6, 3, A, A, C, C, C, 1, 5]
B: [6, 7, 7, 7, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y3, 0, y1, 0, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 4 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1 + y5, y1, y2, 0, y5, y3, 0, 2 y5, y5, y4, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
599 . Coloring, {2, 3, 5, 7, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y4, y8, y5, y6, 0, y7, y8, y9]

  p = s 6 - s 10

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y1, -3 y1 + 7 y5 - 3 y4 + 10 y3 - 3 y2, 0, 3 y5, 3 y4, 0, 3 y3, 3 y5, 3 y2, 3 y5 + 3 y3]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6


 
600 . Coloring, {2, 3, 5, 7, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, C, 4, 5]
B: [6, 7, 7, 7, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y4, y7, y5, 0, y6, y7, y8]

  p = - s 5 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y8 + 13 y7, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 0, 3 y5, 3 y6, 3 y8, 3 y7]

  p = - s 3 - s 4 + s 8 + s 9


 
601 . Coloring, {2, 3, 5, 7, 12}

Ωp(Δ)=0:     p = - 2s 2 + s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, C, 1, 9]
B: [6, 7, 7, 7, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y8, 0, y7, 0, 0, y7, y6, y5, y3, y4, y2, y1]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y7 - y5, y6, y7, y5, y4, 0, 0, y2, y3, y5]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
602 . Coloring, {2, 3, 5, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, C, 1, 5]
B: [6, 7, 7, 7, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y4, 0, y3, 0, y2, y1, y9, y8, 0, y7, y6, y5]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, -y4 + y5, y3, 0, y4, y2, 0, y5, y6, y7, y4]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
603 . Coloring, {2, 3, 5, 8, 10}

R: [7, 8, 8, 6, 3, A, B, B, B, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y6, y7, 0, -2 y7 + 4 y3, -y7 + 2 y3, y5, y4, 0, y3, y2, 0]

  p = s 5 - s 8   p' = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 - y3 - y4 + y5, 2 y1, 0, y1, y2, 0, y3, y4, 0, y5]

  p = - s 5 + s 7   p = - s 5 + s 6


 
604 . Coloring, {2, 3, 5, 8, 11}

R: [7, 8, 8, 6, 3, A, B, B, B, C, 4, 5]
B: [6, 7, 7, 7, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1 - y8 + y7 - y6 - y5 - y4 + y3 + y2, y8, y7, y6, y5, 0, y4, y3, y2]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[y5, y4, y3, 0, 0, y2, y1, 0, y6, -y5 - y4 - y3 - y2 - y1 + 6 y6, 0, y6]

  p' = s 4 - s 7   p = s 4 - s 7


 
605 . Coloring, {2, 3, 5, 8, 12}

R: [7, 8, 8, 6, 3, A, B, B, B, C, 1, 9]
B: [6, 7, 7, 7, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5 - y7, 0, 0, y5 - y7, y4, y5, y6, y7, y2, y3]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y5, y4, 2 y3, 2 y4, y3, y2, 0, 0, y1, 0, 2 y3]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6


 
606 . Coloring, {2, 3, 5, 9, 10}

R: [7, 8, 8, 6, 3, A, B, C, C, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[5 y3, 5 y2, 5 y1, 0, 11 y3 - 5 y2 - 5 y1 - 5 y9 + 11 y8 - 5 y7 - 5 y6 + 11 y5 - 5 y4, 5 y9, 5 y8, 5 y7, 0, 5 y6, 5 y5, 5 y4]

  p = - s 4 - s 5 - s 6 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
607 . Coloring, {2, 3, 5, 9, 11}

R: [7, 8, 8, 6, 3, A, B, C, C, C, 4, 5]
B: [6, 7, 7, 7, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1 - y8 + y7 - y6 - y5 - y4 + y3 + y2, y8, y7, y6, y5, 0, y4, y3, y2]

  p = s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y6, y4, 0, y5, y7, y8, 0]


 
608 . Coloring, {2, 3, 5, 9, 12}

R: [7, 8, 8, 6, 3, A, B, C, C, C, 1, 9]
B: [6, 7, 7, 7, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y1, 0, 5 y4 - 5 y6, 0, 0, 5 y4 - 5 y6, 5 y3, 5 y4, 5 y5, 5 y6, 5 y2, 11 y1 - 15 y4 + 5 y6 + 11 y3 - 5 y5 + 11 y2]

  p = - s 3 + s 9   p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y2, 2 y2, 2 y3, y3, y4, 0, 0, y5, 2 y3, 0]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = s 3 - s 6


 
609 . Coloring, {2, 3, 5, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y6 + y5 + y7 - y8 - y9, y1, y2, y3, y4, y6, y5, 0, y7, y8, y9]

  p = - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, -y1 + y3 + y2 - y7 - y6 + y5 + y4, 0, 0, y3, y2, 0, y7, y6, y5, y4]

  p = s 7 - s 8


 
610 . Coloring, {2, 3, 5, 10, 12}

Ωp(Δ)=0:     p = 6s 2 + s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y1, y2, y3, 0, 0, y3, y4, y5, y6, y9, y7, y8]

  p = - s 7 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, y4, -y6 + y5, y3, 0, 0, y2, -y6 + y5, y1]

  p = s 4 - s 7   p' = s 4 - s 7


 
611 . Coloring, {2, 3, 5, 11, 12}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, C, 4, 9]
B: [6, 7, 7, 7, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y3, 0, y3 + y1 + y7 + y4 - y5 - y6, y2, y1, y7, y4, y5, y6]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, y4, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
612 . Coloring, {2, 3, 6, 7, 8}

R: [7, 8, 8, 6, A, 3, A, B, B, C, 1, 5]
B: [6, 7, 7, 7, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y9, y7, y8]

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, y6, y5, y4, 0, y5, y3, 0, y2, y1, -y6 - y4 - 2 y5 - y3 - y1 + 6 y2, y2]

  p' = s 4 - s 7   p = s 4 - s 7   p' = s 5 - s 8


 
613 . Coloring, {2, 3, 6, 7, 9}

R: [7, 8, 8, 6, A, 3, A, C, C, C, 1, 5]
B: [6, 7, 7, 7, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y5, 0, y2, 0, y1, y5, 2 y2 - y5, y4, 0, y3, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y3, y4, 0, 2 y3, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
614 . Coloring, {2, 3, 6, 7, 10}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 10 4 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2 + y8, y1, y2, 0, y3, y8, y4, y5, 0, y6, y8, y7]

  p' = s 4 - s 9   p = s 4 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 3 y1, -3 y1 + 3 y2, 0, 3 y1, 3 y2, 0, 3 y3, 3 y4, 3 y2, -3 y1 + 7 y2 - 3 y3 - 3 y4]

  p = - s 3 + s 5   p' = - s 3 + s 5   p' = - s 3 + s 7   p = - s 3 + s 7

 ยป SYNC'D !RANK'D


 
615 . Coloring, {2, 3, 6, 7, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, C, 4, 5]
B: [6, 7, 7, 7, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y7, y5, 0, y6, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[3 y2, 3 y3, 3 y1, 0, 0, -3 y3 - 3 y1 + 8 y7 - 3 y5 + 5 y4, -3 y2 + 5 y7 - 3 y6 + 8 y4, 0, 3 y7, 3 y6, 3 y5, 3 y4]

  p = s 2 - s 8   p' = s 2 - s 8

 ยป SYNC'D !RANK'D


 
616 . Coloring, {2, 3, 6, 7, 12}

Ωp(Δ)=0:     p = 2s 2 - s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, C, 1, 9]
B: [6, 7, 7, 7, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y3, 0, y2, 0, 0, y1, y9, y8, y7, y6, y5, y4]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y3, y3 - 2 y4 + y6, y2, y4 + y6, y4, y1, 0, 0, y6, y5, y4]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = - s 4 + s 7


 
617 . Coloring, {2, 3, 6, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, C, 1, 5]
B: [6, 7, 7, 7, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y3, 0, -2 y2 - 2 y3 + 5 y1 + 5 y5 - 4 y6, y1, 4 y2 + 4 y3 - 6 y1 - y4 - 6 y5 + 5 y6, y4, 0, 5 y2 + 5 y3 - 8 y1 - 8 y5 + 6 y6, y5, y6]

  p' = - s 4 + s 7   p' = - s 5 + s 8   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y7, y2, 0, y7, y6, 0, y3, y4, y5, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
618 . Coloring, {2, 3, 6, 8, 10}

R: [7, 8, 8, 6, A, 3, B, B, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 4 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y3, y2, y1, 0, 4 y1 - 2 y7, 2 y1 - y7, y4, y6, 0, y7, y5, 0]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, y1, 2 y1 + y2 + y3 - y4, 0, y2, y3, 0, y4]

  p = - s 4 + s 7   p = - s 4 + s 5   p = - s 4 + s 6


 
619 . Coloring, {2, 3, 6, 8, 11}

R: [7, 8, 8, 6, A, 3, B, B, B, C, 4, 5]
B: [6, 7, 7, 7, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 5 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 5 y8, 5 y5, 5 y6, 5 y7, 5 y3, 5 y4, 0, 5 y1, -5 y8 - 5 y5 + 11 y6 - 5 y7 - 5 y3 - 5 y4 + 11 y1 + 11 y2, 5 y2]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[2 y1, -3 y1 - y2 - y3 - y4 + 6 y5, y1, 0, 0, y2, y3, 0, y5, y4, 0, y5]

  p' = s 4 - s 7   p = s 3 - s 6   p' = s 3 - s 6


 
620 . Coloring, {2, 3, 6, 8, 12}

R: [7, 8, 8, 6, A, 3, B, B, B, C, 1, 9]
B: [6, 7, 7, 7, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y6 - y3 + y4, 0, 0, y6, y5, y3, y4, y6, y2, 2 y6 - y3 + y4]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, y4, 2 y2, y3, y2, y1, 0, 0, y6, 0, 2 y2]

  p' = s 4 - s 7   p = s 4 - s 7


 
621 . Coloring, {2, 3, 6, 9, 10}

R: [7, 8, 8, 6, A, 3, B, C, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[5 y8, 5 y7, 5 y6, 0, 5 y5, 5 y4, 5 y3, 5 y2, 0, 11 y8 - 5 y7 - 5 y6 - 5 y5 - 5 y4 + 11 y3 - 5 y2 + 11 y1 - 5 y9, 5 y1, 5 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y3, 0, y1, y3 + y2 + y4 - y5 - y6, 0, y2, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p' = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
622 . Coloring, {2, 3, 6, 9, 11}

R: [7, 8, 8, 6, A, 3, B, C, C, C, 4, 5]
B: [6, 7, 7, 7, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, 0, 2 y3, y6, y7, 0]

  p = - s 5 + s 8


 
623 . Coloring, {2, 3, 6, 9, 12}

R: [7, 8, 8, 6, A, 3, B, C, C, C, 1, 9]
B: [6, 7, 7, 7, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y4, 0, 5 y3, 0, 0, 5 y7, 5 y2, 5 y1, 11 y4 - 5 y3 - 10 y7 + 11 y2 - 5 y1 + 11 y6 - 5 y5, 5 y7, 5 y6, 5 y5]

  p = s 4 - s 6 - s 7 + s 9   p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y2 - y3, y2, 2 y3, y3, y4, 0, 0, y5, 2 y3, 0]

  p = s 3 - s 6   p' = s 3 - s 6   p' = s 4 - s 7


 
624 . Coloring, {2, 3, 6, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 10 vs 10 5 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y10, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y5, 0, y4, 0, 0, y3, y5 + y4, 0, y2, y1, y4, -y4 - y3 + y2 + y1]

  p' = s 6 - s 7   p' = s 5 - s 7   p = s 5 - s 8


 
625 . Coloring, {2, 3, 6, 10, 12}

Ωp(Δ)=0:     p = - 2s 2 + 5s 3 + 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 + y6, y2, 0, 0, y6, y3, y4, y5, y6, y7, y8]

  p' = - s 6 + s 9   p = - s 6 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, y2, y3, y6, y1, 0, 0, y5, y6, y7]

  p = s 3 - s 8

 ยป SYNC'D !RANK'D


 
626 . Coloring, {2, 3, 6, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 5s 3 - 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, C, 4, 9]
B: [6, 7, 7, 7, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y7, y8, 0, y5, y6, y4, y3, y6, y1, y2]

  p = - s 2 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, y2, y3, y4, 0, 0, y5, y6, y6]

  p' = s 4 - s 7   p' = s 5 - s 8   p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
627 . Coloring, {2, 3, 7, 8, 9}

R: [7, 8, 8, 6, A, A, A, B, C, C, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y5, y6, 2 y5, 0, y7, y2, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y5, y3, y1, 0, y5, y2, 0, y3, 0, y4, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
628 . Coloring, {2, 3, 7, 8, 10}

Ωp(Δ)=0:     p = - 2s 2 + 5s 3 + 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y1 - 3 y2 - y5 + y6 + y4 - y3, 0, 0, 2 y2, y2, y5, y6, 0, y4, y3, 0]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -3 y2 - 3 y1 - 3 y3 + 5 y4 - 3 y6 + 5 y5, 3 y2, 0, 3 y1, 3 y3, 0, 3 y4, 0, 3 y6, 3 y5]

  p = s 3 + s 4 - s 6 - s 7

 ยป SYNC'D !RANK'D


 
629 . Coloring, {2, 3, 7, 8, 11}

Ωp(Δ)=0:     p = 2s 2 + 5s 3 - 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[-y1 - y2 - y3 - y4 + 6 y5 - y6, y1, y2, 0, 0, y3, y4, 0, y5, 0, y6, y5]

  p' = - s 2 + s 7   p = - s 2 + s 7


 
630 . Coloring, {2, 3, 7, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y7, 2 y2, y6, y5, y4, y3]

  p = s 2 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y3, y1, y4, y5, y3, y2, 0, 0, 0, y6, 2 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
631 . Coloring, {2, 3, 7, 9, 10}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y6, y1, 0, 0, y5, y6, y4, y2, 0, y3, 0, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, y1, y2, 0, y3, y5, 0, 2 y1 - 2 y3, 0, y4, 2 y3]

  p = s 3 - s 6   p' = s 3 - s 6


 
632 . Coloring, {2, 3, 7, 9, 11}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y4, y1, y2, y4, 2 y4, 0, y3, 0, y5]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y5, y2, 0, 0, y3, y4, 0, y5, 0, y6, 0]

  p = - s 2 + s 7

 ยป SYNC'D !RANK'D


 
633 . Coloring, {2, 3, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, 0, 0, 0, y4, y3, 2 y4, y1, y2, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y1, y3, y4, 2 y1, y1, y2, 0, 0, 0, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

 ยป SYNC'D !RANK'D


 
634 . Coloring, {2, 3, 7, 10, 11}

R: [7, 8, 8, 6, A, A, A, C, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, y4, y8, y5, 0, y6, y8, y7]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[-5 y3 - 5 y1 - 5 y2 + 11 y4 - 5 y5 + 11 y6, 0, 5 y3, 0, 0, 5 y1, 5 y2, 0, 5 y4, 0, 5 y5, 5 y6]

  p = - s - s 2 + s 6 + s 7


 
635 . Coloring, {2, 3, 7, 10, 12}

R: [7, 8, 8, 6, A, A, A, C, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y7, y3, y4, y5, y6, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y6, 0, 0, 0, y5, 3 y4]

  p = - s 4 + s 7


 
636 . Coloring, {2, 3, 7, 11, 12}

R: [7, 8, 8, 6, A, A, A, C, B, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, 2 y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 2 y6, y2, 0, y3, y4, y5, 0, 0, 0, y7, y6]

  p = - s 3 + s 8


 
637 . Coloring, {2, 3, 8, 9, 10}

R: [7, 8, 8, 6, A, A, B, B, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y3, 0, 0, y2, y8, y4, y5, 0, y7, y6, y8]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
638 . Coloring, {2, 3, 8, 9, 11}

R: [7, 8, 8, 6, A, A, B, B, C, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, 0, y6, y7, y8, y9]


 
639 . Coloring, {2, 3, 8, 9, 12}

R: [7, 8, 8, 6, A, A, B, B, C, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-y1 - y4 + 2 y5, 0, 0, 0, 0, y2 + y3 - y5, y1, 2 y2 + 2 y3 - 2 y5, y2, y3, y4, y5]

  p = s 3 - s 6   p' = s 3 - s 6   p' = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y4, y4, y6, y3, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
640 . Coloring, {2, 3, 8, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, y3, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-y1 + y4 + y5 - y6 - y2 + y3, 0, y1, 0, 0, y4, y5, 0, y6, y2, 0, y3]

  p = - s 6 + s 7


 
641 . Coloring, {2, 3, 8, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y5, y6, y7, 2 y5, y3, y4, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, 2 y3, y2, y3, y4, 0, 0, y5, 0, y6]

  p = - s 2 + s 7


 
642 . Coloring, {2, 3, 8, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, y6, y5, 2 y5, y4, y3, y2, y1]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y3 - 2 y6, y2, y1, 0, y3, y6, y5, 0, 0, y4, 0, 2 y3 - 2 y6]

  p' = - s 4 + s 7   p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
643 . Coloring, {2, 3, 9, 10, 11}

Ωp(Δ)=0:     p = - 6s 2 + 5s 3 - 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y9, y8, y6, y7, y5, 0, y4, y3, y2]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y3 - y6 - y7 + y4 + y5, 0, y1, 0, 0, y2, y3, 0, y6, y7, y4, y5]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
644 . Coloring, {2, 3, 9, 10, 12}

Ωp(Δ)=0:     p = - 9s 3 + 2s 4 + 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y8, 5 y7, 0, 0, 0, 5 y6, 5 y5, 5 y4, 5 y3, 5 y2, 5 y1, 11 y8 - 5 y7 - 5 y6 + 11 y5 - 5 y4 - 5 y3 - 5 y2 + 11 y1]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y6, y7, y3, y4, y5, 0, 0, y2, 2 y4, y1]

  p = - s 3 + s 8


 
645 . Coloring, {2, 3, 9, 11, 12}

Ωp(Δ)=0:     p = 9s 3 + 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y1, y7, 2 y7, y6, y5, y4, y3]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y7, y4, y5, 0, 0, y6, y7, 0]

  p = - s 5 + s 8


 
646 . Coloring, {2, 3, 10, 11, 12}

R: [7, 8, 8, 6, A, A, B, C, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y9, 0, y8, 0, y7, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y8, 0, y7, 0, y6, y5, y4, 0, 0, y3, y2, y1]


 
647 . Coloring, {2, 4, 5, 6, 7}

R: [7, 8, 7, 7, 3, 3, A, C, B, C, 1, 5]
B: [6, 7, 8, 6, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, 0, y3, y4, 0, y2, y4, y1]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -3 y1 + 8 y4 - 3 y6 + 5 y7, 0, -3 y2 - 3 y3 + 5 y4 - 3 y5 + 8 y7, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
648 . Coloring, {2, 4, 5, 6, 8}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, C, 1, 5]
B: [6, 7, 8, 6, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, 0, y6, 2 y5]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y6, 0, 10 y6 + 10 y5 + 10 y4 - 22 y3 + 10 y2 - 22 y1, 0, y5, y4, 5 y6 + 5 y5 + 5 y4 - 11 y3 + 5 y2 - 11 y1, y3, y2, 0, y1]

  p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7


 
649 . Coloring, {2, 4, 5, 6, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, C, 1, 5]
B: [6, 7, 8, 6, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, 0, y6, 0, y1, y2, y3, 2 y3, y4, y7, 0]

  p = - s 5 + s 8


 
650 . Coloring, {2, 4, 5, 6, 10}

R: [7, 8, 7, 7, 3, 3, B, C, B, 2, 1, 5]
B: [6, 7, 8, 6, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, y7, y6, 0, y5, 0, y4, y3, 0, 0, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y4, y5, y5, y2, y3, y7, y6]

  p = s 6 - s 8


 
651 . Coloring, {2, 4, 5, 6, 11}

R: [7, 8, 7, 7, 3, 3, B, C, B, C, 4, 5]
B: [6, 7, 8, 6, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y1, y2, 0, y5, y6, 0, 0, y7, y4]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = s 5 + s 6 - s 8 - s 9


 
652 . Coloring, {2, 4, 5, 6, 12}

R: [7, 8, 7, 7, 3, 3, B, C, B, C, 1, 9]
B: [6, 7, 8, 6, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, 2 y3, 0, 0, 0, y2, y3, y6, 0, y5, y4]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, y7 + y6, y3, y4, y7, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = s 5 - s 8


 
653 . Coloring, {2, 4, 5, 7, 8}

R: [7, 8, 7, 7, 3, A, A, B, B, C, 1, 5]
B: [6, 7, 8, 6, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   10

See Matrix
 

[0, y7, y8, y6, 0, y5, y4, y2, y3, y1, y9, y10]


 
654 . Coloring, {2, 4, 5, 7, 9}

R: [7, 8, 7, 7, 3, A, A, C, C, C, 1, 5]
B: [6, 7, 8, 6, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y4, 0, y2, 0, y1, 0, y3, y4, 0, y5, 0, y6]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y7, y7, y8, 0]

  p = s 4 - s 9


 
655 . Coloring, {2, 4, 5, 7, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y7, y6, y5, 0, y4, 0, y3, y2, 0, y1, y8, y9]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -3 y1 - 3 y6 - 3 y3 - 3 y2 + 10 y5 - 3 y4, 3 y1, 0, 3 y6, 3 y3, 3 y2, -3 y3 + 3 y5, 3 y3, 3 y4, 3 y5]

  p' = s 3 - s 8   p = - s 2 + s 7   p' = - s 2 + s 7


 
656 . Coloring, {2, 4, 5, 7, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, C, 4, 5]
B: [6, 7, 8, 6, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y1, y5, 0, y3, y4, 0, y6, y4, y7]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[3 y2, -3 y2 - 3 y1 - 3 y3 - 3 y4 - 3 y5 + 13 y6 - 3 y7 - 3 y8 + 13 y9, 3 y1, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 3 y9]

  p = s 4 + s 5 - s 9 - s 10


 
657 . Coloring, {2, 4, 5, 7, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, C, 1, 9]
B: [6, 7, 8, 6, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, y6, y7, y8, y1, y2, y10, y3, 0, y4, y5, y9]


 
658 . Coloring, {2, 4, 5, 8, 9}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, C, 1, 5]
B: [6, 7, 8, 6, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}, {2, 7, 10}}   

See Matrix
 

[0, 3 y1, 3 y8, 5 y1 - 3 y8 - 3 y2 + 5 y3 - 3 y4 - 3 y5 + 5 y6 - 3 y7 - 3 y9, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y9]

  p = - s - s 2 - s 3 + s 8 + s 9 + s 10


 
659 . Coloring, {2, 4, 5, 8, 10}

R: [7, 8, 7, 7, 3, A, B, B, B, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 2 y2 - 3 y6, 0, 2 y6, 0, y3, y4, 0, y6, y5, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y5, 2 y2, 0, -4 y2 + 2 y1, y2, y4, y3, y1, 0, -y5 - 7 y2 + y4 + y3 + 3 y1]

  p' = s 6 - s 7   p' = s 5 - s 7   p = s 5 - s 8


 
660 . Coloring, {2, 4, 5, 8, 11}

R: [7, 8, 7, 7, 3, A, B, B, B, C, 4, 5]
B: [6, 7, 8, 6, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y2, y3, 0, y1, y6, 0, y6, y7, y5]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-5 y1 + 5 y2 + 5 y7 + 5 y6 - 5 y5, 6 y2 - 5 y3 + 6 y7 + 6 y6 - 5 y4, 5 y1, 0, 0, 5 y2, 5 y3, 5 y7, 5 y6, 5 y4, 0, 5 y5]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
661 . Coloring, {2, 4, 5, 8, 12}

R: [7, 8, 7, 7, 3, A, B, B, B, C, 1, 9]
B: [6, 7, 8, 6, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y6, 0, 0, 0, y2, y6, y5, y6, y3, y4]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, y5, y8, y9, 0, y6, 0, y7]


 
662 . Coloring, {2, 4, 5, 9, 10}

R: [7, 8, 7, 7, 3, A, B, C, C, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y2, y7, y1, 0, y6, 0, y5, y4, 0, y3, y8, y9]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y6, y4, 0, y5, y3, y2, y9, y1, y7, y8]


 
663 . Coloring, {2, 4, 5, 9, 11}

R: [7, 8, 7, 7, 3, A, B, C, C, C, 4, 5]
B: [6, 7, 8, 6, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y5, 0, y4, y3, 0, y3, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[5 y1 - 3 y2 - 3 y3 + 5 y4 - 3 y5 - 3 y6 + 5 y7 - 3 y8, 3 y1, 3 y2, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
664 . Coloring, {2, 4, 5, 9, 12}

R: [7, 8, 7, 7, 3, A, B, C, C, C, 1, 9]
B: [6, 7, 8, 6, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-y2 - y1 + 2 y4 - y3, 0, y2, 0, 0, 0, y1, y2, -2 y2 + y4, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, y1 + y2 - y3 + y4 - y7 + y8 - y5 + y6, y1, y2, y3, y4, y7, y8, 0, y5, y6, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
665 . Coloring, {2, 4, 5, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y8, y9, y7, y6, 0, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[7 y2, 0, 7 y1, 0, 0, 7 y8, 7 y7, 7 y6, 7 y5, 7 y4, 7 y3, 9 y2 + 9 y1 + 9 y8 - 7 y7 + 9 y6 - 7 y5 - 7 y4 + 9 y3]

  p = s 3 + s 4 - s 8 - s 9


 
666 . Coloring, {2, 4, 5, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, y5, 0, 0, 0, y8, y6, y7, y5, y3, y4]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 3 y4, 3 y3, 3 y2, 3 y1, -3 y2 + 3 y5, -3 y4 - 3 y3 - 3 y1 + 7 y5 - 3 y6, 0, 3 y5, 3 y6, 3 y5]

  p' = - s 2 + s 7   p' = - s 3 + s 8   p = - s 2 + s 7


 
667 . Coloring, {2, 4, 5, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, C, 4, 9]
B: [6, 7, 8, 6, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y6, y5, 0, 0, y4, y6, y3, y6, y2, y1]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[-3 y4 - 3 y2 - 3 y1 + 7 y5 - 3 y7, 3 y5 - 3 y6, 3 y4, 0, 3 y3, 3 y2, -3 y3 + 3 y5, 3 y1, 0, 3 y5, 3 y7, 3 y6]

  p' = s 3 - s 8   p = s 3 - s 8   p' = s 4 - s 9


 
668 . Coloring, {2, 4, 6, 7, 8}

R: [7, 8, 7, 7, A, 3, A, B, B, C, 1, 5]
B: [6, 7, 8, 6, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y7, y2, 0, y4, y5, y6]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y2, 3 y7 + 3 y6 - 3 y3, 3 y1, 0, -3 y2 + 10 y7 + 10 y6 - 3 y1 - 3 y8 - 3 y5 - 3 y4, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3]

  p = s 3 - s 9   p' = - s 3 + s 9


 
669 . Coloring, {2, 4, 6, 7, 9}

R: [7, 8, 7, 7, A, 3, A, C, C, C, 1, 5]
B: [6, 7, 8, 6, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, y3, 0, y1, 0, y2, y3, 0, y4, 0, y5]

  p' = - s 3 + s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y7, y8, y6, 0, y5, y4, y3, 2 y8, y2, y1, 0]

  p = - s 3 + s 9


 
670 . Coloring, {2, 4, 6, 7, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y6, y7, y5, 0, y4, 0, y2, y3, 0, y1, y5, y8]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y8, y7, 0, y6, y8, y5, y4, y3, y2, y1]

  p = s 7 - s 9


 
671 . Coloring, {2, 4, 6, 7, 11}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, C, 4, 5]
B: [6, 7, 8, 6, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, y4, 0, y3, y6, 0, y2, y6, y1]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[11 y1 + 11 y2 + 11 y3 - 2 y4 - 2 y5 - 39 y6 - 2 y8 + 11 y7, 2 y1, 2 y2, 0, 0, 2 y3, 2 y4, 2 y5, 2 y6, 2 y8, 2 y7, 3 y1 + 3 y2 + 3 y3 - 11 y6 + 3 y7]

  p' = - s 3 + s 9   p = - s 3 + s 9


 
672 . Coloring, {2, 4, 6, 7, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, C, 1, 9]
B: [6, 7, 8, 6, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y6, 0, 0, 0, y5, y6, y3, y2, y4, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y8, y9, y10, y1, y2, y3, y4, 0, y5, y6, y7]


 
673 . Coloring, {2, 4, 6, 8, 9}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, C, 1, 5]
B: [6, 7, 8, 6, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[-7 y4 - 6 y3 + 4 y2 + 5 y1, 0, y4, 0, 5 y4 + 5 y3 - 2 y2 - 4 y1, 0, y3, y4, 0, -8 y4 - 8 y3 + 5 y2 + 6 y1, y2, y1]

  p' = - s 4 + s 7   p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y8, y9, 0, y10, y4, y5, y6, y7, y3, y2]


 
674 . Coloring, {2, 4, 6, 8, 10}

R: [7, 8, 7, 7, A, 3, B, B, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y3, 0, y4, y5, 0, y6, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, 2 y2 + 2 y3 - 2 y4, y1, y2 + y3 - y4, y2, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8


 
675 . Coloring, {2, 4, 6, 8, 11}

R: [7, 8, 7, 7, A, 3, B, B, B, C, 4, 5]
B: [6, 7, 8, 6, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 5 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 2 y2, 7 y2 - 2 y3 + 9 y1 - 11 y4, -2 y2 + 2 y3 - 2 y1 + 2 y4, 0, 2 y1, 2 y2, 0, 7 y2 - 2 y3 + 7 y1 - 9 y4, 2 y3, 2 y4]

  p = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[6 y5, 3 y4, 3 y5, 0, 0, 6 y3, 3 y2, 3 y3, 3 y5 - 3 y3 + 3 y1, -3 y4 + 4 y5 - 3 y2 - 6 y3 + 10 y1, 0, 3 y1]

  p' = s 4 - s 7   p = - s 3 + s 6   p = - s 3 + s 9   p' = - s 3 + s 6


 
676 . Coloring, {2, 4, 6, 8, 12}

R: [7, 8, 7, 7, A, 3, B, B, B, C, 1, 9]
B: [6, 7, 8, 6, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, 0, y4, y2, y5, y2, y3, y6]

  p' = s 4 - s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 3 y6, 3 y5, 3 y4, 3 y3, 3 y1, 3 y2, 3 y5 - 3 y3 + 3 y7, 0, -3 y6 + 10 y5 - 3 y4 - 3 y1 - 3 y2 + 10 y7, 0, 3 y7]

  p = s 3 + s 5 - s 6 - s 8   p' = s 3 + s 5 - s 6 - s 8


 
677 . Coloring, {2, 4, 6, 9, 10}

Ωp(Δ)=0:     p = s 4 - 16s 8   p' = s 4 - 8s 7   p' = s 5 - 4s 7   p' = s 6 - 2s 7

R: [7, 8, 7, 7, A, 3, B, C, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
4 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[y1, y2, -y1 + y2 + y3 - y4 + y5 + y6 - y7 + y8, 0, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1 + 2 y4 - y3 - y2 - y5 + y7 + y6, 0, y1, y4, y3, y2, y5, y7, y6]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
678 . Coloring, {2, 4, 6, 9, 11}

R: [7, 8, 7, 7, A, 3, B, C, C, C, 4, 5]
B: [6, 7, 8, 6, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -y2 - 2 y1 + 2 y3 + y4, y2 + y1 - y3, 2 y1 - y4, 0, y1, -y2 - 2 y1 + 2 y3 + y4, 0, y2, y3, y4]

  p' = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 2 + s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y5, y4, y3, 0, 0, y2, y1, y8, 2 y3, y7, y6, 0]

  p = s 6 - s 9


 
679 . Coloring, {2, 4, 6, 9, 12}

R: [7, 8, 7, 7, A, 3, B, C, C, C, 1, 9]
B: [6, 7, 8, 6, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, 0, y2, y4, y3, y4, -y1 - y2 + 3 y4 + 2 y3, 2 y4 + y3]

  p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8   p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y9, y8, 0]


 
680 . Coloring, {2, 4, 6, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, -3 y2 + 7 y5 - 3 y1 - 3 y4 - 3 y6, -3 y3 + 3 y5, 3 y3, 3 y2, 0, 3 y5, 3 y1, 0, 3 y4, 3 y5, 3 y6]

  p' = s 2 - s 7   p' = s 3 - s 8   p = s 2 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 + 2 y2 - y3 - y7 - y4 + y5 + y6, 0, y2, 0, 0, y1, y2, y3, y7, y4, y5, y6]

  p = - s 7 + s 8   p = - s 7 + s 9


 
681 . Coloring, {2, 4, 6, 10, 12}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, y5, y7, 0, 0, 0, y2, y3, y1, y7, y8, y6]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3, y2, -y2 + y5 + y4, y1, -y3 + y5 + y4 - y1, y5 + y4 - y6, 0, y6, y5, y4]

  p' = - s 2 + s 4 - s 6 + s 8   p' = - s 2 + s 3 - s 6 + s 7   p = s 2 - s 3 + s 6 - s 7


 
682 . Coloring, {2, 4, 6, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, C, 4, 9]
B: [6, 7, 8, 6, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y1, 0, 0, y6, y5, y4, y5, y3, y2]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y10, y9, y8, 0, y7, y6, y5, y4, 0, y3, y2, y1]


 
683 . Coloring, {2, 4, 7, 8, 9}

R: [7, 8, 7, 7, A, A, A, B, C, C, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y4, 0, 0, 0, y2, 0, y3, y1, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y1, y2, y5, 0, y6, y7, y8, y9, 0, y3, y4]


 
684 . Coloring, {2, 4, 7, 8, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y4, y5, 0, 0, y3, 0, y1, y2, 0, y6, y7, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, 0, y7, y8]


 
685 . Coloring, {2, 4, 7, 8, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y1, 0, y2, y3, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y4, y3, y2, 0, 0, y1, y9, y8, y7, 0, y6, y5]


 
686 . Coloring, {2, 4, 7, 8, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y8, y9, y7, y5, y6, y4, y3, 0, 0, y1, y2]


 
687 . Coloring, {2, 4, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, y5, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y7, y1, 0, y2, y3, y4, y5, 0, y6, 2 y3]

  p = - s 3 + s 8


 
688 . Coloring, {2, 4, 7, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y2, y1, 0, y4, y2, 0, y3, 0, y5]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y5, y6, 0, 0, y2, y3, y4, y5, 0, y7, 0]

  p = - s 3 + s 8


 
689 . Coloring, {2, 4, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, 0, 0, 0, 0, y1, y2, y5, y4, 0, y3]

  p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y7, y5, y6, y7, y4, y2, y3, 0, 0, y1, 0]

  p = s 3 - s 8


 
690 . Coloring, {2, 4, 7, 10, 11}

R: [7, 8, 7, 7, A, A, A, C, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[-5 y1 - 5 y2 - 5 y3 - 5 y7 + 11 y6 - 5 y4 + 11 y5, 0, 5 y1, 0, 0, 5 y2, 5 y3, 5 y7, 5 y6, 0, 5 y4, 5 y5]

  p = - s 2 - s 3 + s 7 + s 8


 
691 . Coloring, {2, 4, 7, 10, 12}

R: [7, 8, 7, 7, A, A, A, C, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y7, y5, 0, 0, y6, 3 y7]

  p = - s 3 + s 8


 
692 . Coloring, {2, 4, 7, 11, 12}

R: [7, 8, 7, 7, A, A, A, C, B, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, 0, y1, y2, y6, y4, y5, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, y3, y4, 2 y3 - 3 y7, y5, 0, 0, y6, y7]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
693 . Coloring, {2, 4, 8, 9, 10}

R: [7, 8, 7, 7, A, A, B, B, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y8, 0, y6, y7, 0, y3, y4, y5]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
694 . Coloring, {2, 4, 8, 9, 11}

R: [7, 8, 7, 7, A, A, B, B, C, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y1, 2 y2, 0, 7 y1 - 27 y2 - 2 y3 + 16 y4, 2 y3, 0, 3 y1 - 7 y2 + 4 y4, 2 y4, 4 y1 - 16 y2 + 10 y4]

  p' = - s 2 + s 5   p = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}, {2, 7, 10}}   

See Matrix
 

[11 y9 - 5 y8 - 5 y6 + 11 y7 - 5 y5 - 5 y4 + 11 y3 - 5 y2 - 5 y1, 5 y9, 5 y8, 0, 0, 5 y6, 5 y7, 5 y5, 5 y4, 5 y3, 5 y2, 5 y1]

  p = - s - s 2 - s 3 + s 8 + s 9 + s 10


 
695 . Coloring, {2, 4, 8, 9, 12}

R: [7, 8, 7, 7, A, A, B, B, C, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[7 y5, 0, 0, 0, 0, 0, 7 y4, 7 y3, 7 y2, 14 y3, -7 y5 - 7 y4 + 11 y3 + 9 y2 + 9 y1, 7 y1]

  p = s 2 + s 3 - s 5 - s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 5 y1, 11 y1 - 5 y9 - 5 y5 - 5 y6 + 11 y7 - 5 y8 + 11 y2 - 5 y3 - 5 y4, 5 y9, 5 y5, 5 y6, 5 y7, 5 y8, 0, 5 y2, 5 y3, 5 y4]

  p = - s 4 - s 5 - s 6 + s 8 + s 9 + s 10


 
696 . Coloring, {2, 4, 8, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, 0, y4, y5, 0, y6, y7, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y5, 0, y6, 0, 0, -2 y5 + 2 y2, y5, y4, y3, y2, 0, y1]

  p' = s 5 - s 7   p = s 5 - s 7


 
697 . Coloring, {2, 4, 8, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y5, y6, y4, y4, y3, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y6, 2 y4, y5, -2 y4 + 2 y2, y4, y3, 0, y2, 0, y1]

  p' = - s 3 + s 7   p = - s 3 + s 7


 
698 . Coloring, {2, 4, 8, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y6, y4, y5, 2 y4, y2, y3]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[5 y2, 5 y1, -5 y2 + 11 y1 - 5 y5 - 5 y6 + 11 y7 - 5 y8 + 11 y3 - 5 y4, 0, 5 y5, 5 y6, 5 y7, 5 y8, 0, 5 y3, 0, 5 y4]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9


 
699 . Coloring, {2, 4, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, 9 y1 - 7 y2 + 9 y3 - 7 y4 - 7 y5 + 9 y6 - 7 y7, 0, 7 y1, 7 y2, 0, 7 y3, 7 y4, 0, 7 y5, 7 y6, 7 y7]

  p = s + s 2 + s 3 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y5, y6, y4, y3, y7, y8, y9]


 
700 . Coloring, {2, 4, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[7 y7, 7 y6, 0, 0, 0, 0, 7 y5, 7 y4, 7 y2, 7 y3, 7 y1, 9 y7 - 7 y6 + 9 y5 - 7 y4 - 7 y2 - 7 y3 + 9 y1]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
701 . Coloring, {2, 4, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, 7 y5, 0, 0, 7 y4, 7 y3, 7 y2, 14 y3, 7 y1, 9 y5 + 9 y4 - 21 y3 - 7 y2 + 9 y1]

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[5 y8, 5 y7, 5 y6, 0, 5 y5, 5 y4, 5 y3, 5 y2, 0, 5 y1, -5 y8 + 11 y7 - 5 y6 - 5 y5 - 5 y4 + 11 y3 - 5 y2 + 11 y1, 0 ]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
702 . Coloring, {2, 4, 10, 11, 12}

R: [7, 8, 7, 7, A, A, B, C, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, 0, y8, y7, y6, y5, y4, y3]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
703 . Coloring, {2, 5, 6, 7, 8}

R: [7, 8, 7, 6, 3, 3, A, B, B, C, 1, 5]
B: [6, 7, 8, 7, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y3, 0, y2, 0, y1, y8, y7, y8, 0, y6, y5, y4]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -5 y1 + 6 y6 - 5 y7 + 11 y5 - 5 y2 - 5 y3 + 11 y4, 0, 5 y1, 0, 5 y6, 5 y7, 5 y6, 5 y5, 5 y2, 5 y3, 5 y4]

  p = s 4 - s 6 - s 7 + s 9   p = - s 4 - s 5 + s 7 + s 8


 
704 . Coloring, {2, 5, 6, 7, 9}

R: [7, 8, 7, 6, 3, 3, A, C, C, C, 1, 5]
B: [6, 7, 8, 7, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y3, 0, y4, 0, y5, y3, y2, y3, 0, y1, 0, y6]

  p = s 2 - s 7   p' = s 2 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, 0, y5, 0, y4, y3, y4, 2 y4, y2, y1, 0]

  p' = s 4 - s 7   p = s 4 - s 7


 
705 . Coloring, {2, 5, 6, 7, 10}

Ωp(Δ)=0:     p = - s + s 2 + s 4 - 4s 5 + 8s 7 - 16s 8

R: [7, 8, 7, 6, 3, 3, A, C, B, 2, 1, 5]
B: [6, 7, 8, 7, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y4, y1, y2, 0, y3, y8, y5, y6, 0, y7, y8, y9]

  p = - s 3 + s 10

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y4, -y1 - y4 + 2 y5 + 2 y3 - y2, y4, y5, y3, y2, -y4 + y5 + y3]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7


 
706 . Coloring, {2, 5, 6, 7, 11}

R: [7, 8, 7, 6, 3, 3, A, C, B, C, 4, 5]
B: [6, 7, 8, 7, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y3, y2, y4, y5, y7, 0, y8, y7, y6]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[-3 y2 - 3 y3 + 8 y4 - 3 y5 + 5 y7, -3 y1 + 5 y4 - 3 y6 + 8 y7, 0, 0, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = s 2 - s 8


 
707 . Coloring, {2, 5, 6, 7, 12}

R: [7, 8, 7, 6, 3, 3, A, C, B, C, 1, 9]
B: [6, 7, 8, 7, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y3, 0, y4, 0, 0, y6, y5, y6, y7, y8, y1, y2]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, 0, y5, y4, y7, y3, y7, 0, y2, y1, y7]

  p' = s 5 - s 8   p = s 5 - s 8


 
708 . Coloring, {2, 5, 6, 8, 9}

R: [7, 8, 7, 6, 3, 3, B, B, C, C, 1, 5]
B: [6, 7, 8, 7, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y3, 0, y1, 0, y2, y5, y4, y5, 0, 0, y6, 3 y5]

  p' = s 4 - s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, 0, y5, 0, y3, y4, y3, y1, y2, y8, y7]

  p = - s 6 + s 9


 
709 . Coloring, {2, 5, 6, 8, 10}

R: [7, 8, 7, 6, 3, 3, B, B, B, 2, 1, 5]
B: [6, 7, 8, 7, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y5, 4 y4 - 6 y2, y4, 0, 4 y4 - 6 y2, 2 y4 - 3 y2, y3, y2, 0, 0, y1, 0]

  p' = s 4 - s 7   p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y2, 0, y2, y1, y2, y5, y4, 0, y3]

  p' = s 4 - s 6   p = s 4 - s 6


 
710 . Coloring, {2, 5, 6, 8, 11}

R: [7, 8, 7, 6, 3, 3, B, B, B, C, 4, 5]
B: [6, 7, 8, 7, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y5, y4, y1, y2, y3, y6, 0, 0, y7, 2 y6]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[10 y1 + 10 y3 + 10 y4 - 22 y5 + 10 y2 - 22 y6, y1, 0, 0, 0, y3, y4, 5 y1 + 5 y3 + 5 y4 - 11 y5 + 5 y2 - 11 y6, y5, y2, 0, y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8


 
711 . Coloring, {2, 5, 6, 8, 12}

R: [7, 8, 7, 6, 3, 3, B, B, B, C, 1, 9]
B: [6, 7, 8, 7, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y2, 0, y1, 0, 0, y4, y3, y4, 2 y1 - 2 y4, 0, y5, 2 y4]

  p = s 3 - s 6   p' = s 3 - s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, 0, 2 y3, y5, y3, y4, y3, 0, y2, 0, y1]

  p' = s 4 - s 7   p = s 4 - s 7


 
712 . Coloring, {2, 5, 6, 9, 10}

R: [7, 8, 7, 6, 3, 3, B, C, C, 2, 1, 5]
B: [6, 7, 8, 7, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y5, y7, 0, y6, y5, y4, y3, 0, 0, y2, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y5, y1, y5, y3, y4, y7, y6]

  p = - s 2 + s 8


 
713 . Coloring, {2, 5, 6, 9, 11}

R: [7, 8, 7, 6, 3, 3, B, C, C, C, 4, 5]
B: [6, 7, 8, 7, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, 0, y7, y8]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, 0, 0, 0, y4, y3, y2, 2 y2, y1, y7, 0]

  p = s 5 - s 8


 
714 . Coloring, {2, 5, 6, 9, 12}

R: [7, 8, 7, 6, 3, 3, B, C, C, C, 1, 9]
B: [6, 7, 8, 7, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y2, 0, 2 y1, 0, 0, 7 y2 + 7 y1 + 7 y3 - 9 y4 + 7 y5 - 9 y6, 2 y3, 7 y2 + 7 y1 + 7 y3 - 9 y4 + 7 y5 - 9 y6, 2 y4, 0, 2 y5, 2 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, y4, y3, y4, 0, y6, y5, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
715 . Coloring, {2, 5, 6, 10, 11}

R: [7, 8, 7, 6, 3, 3, B, C, B, 2, 4, 5]
B: [6, 7, 8, 7, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y9, y8, y7, y6, y5, y4, y3, 0, 0, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y6, 2 y5, y5, y4, y3, y2, y1]

  p = s 6 - s 8


 
716 . Coloring, {2, 5, 6, 10, 12}

R: [7, 8, 7, 6, 3, 3, B, C, B, 2, 1, 9]
B: [6, 7, 8, 7, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y3, y2, 0, 0, y3, y4, 2 y2 - 3 y3, y5, 0, y7, y6]

  p' = - s 5 + s 8   p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, y3, y4, y3, 0, y7, y6, y5]

  p = - s 5 + s 8


 
717 . Coloring, {2, 5, 6, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, C, B, C, 4, 9]
B: [6, 7, 8, 7, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y4, y5, 0, y3, y6, y1, y2, 0, y7, y8]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y7 + y6, y3, y4, y7, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
718 . Coloring, {2, 5, 7, 8, 9}

R: [7, 8, 7, 6, 3, A, A, B, C, C, 1, 5]
B: [6, 7, 8, 7, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y3, y5, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, y5, y4, 0, y6 - y5, y3, y2, y1, y6 - y5, y8, y7 ]

  p' = s 6 - s 9   p = s 6 - s 9


 
719 . Coloring, {2, 5, 7, 8, 10}

R: [7, 8, 7, 6, 3, A, A, B, B, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 4 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y8, y7, y6, 0, 2 y5, y5, y4, y3, 0, y2, y1, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 5 y3 - 2 y4, 2 y3, 0, 2 y2, 2 y3, 5 y3 - 2 y1 - 4 y2, 2 y1, 2 y2, 2 y3, 2 y4]

  p = - s 4 + s 7   p = - s 4 + s 9   p = - s 4 + s 8   p = - s 4 + s 5   p = - s 4 + s 6


 
720 . Coloring, {2, 5, 7, 8, 11}

R: [7, 8, 7, 6, 3, A, A, B, B, C, 4, 5]
B: [6, 7, 8, 7, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y9, y8, y7, y6, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   10

See Matrix
 

[y2 - y1 + y3 - y4 + y5 + y6 - y7 + y8 - y9, y2, y1, 0, 0, y3, y4, y5, y6, y7, y8, y9]

  p = - s 9 + s 10


 
721 . Coloring, {2, 5, 7, 8, 12}

R: [7, 8, 7, 6, 3, A, A, B, B, C, 1, 9]
B: [6, 7, 8, 7, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y3, y2, y3, y4, y5, y7, y6]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y2, y3, y10, y9, y7, y8, 0, y6, y5, y4]


 
722 . Coloring, {2, 5, 7, 9, 10}

R: [7, 8, 7, 6, 3, A, A, C, C, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[2 y7, y2, y1, 0, y8, y7, y6, y5, 0, y4, 0, y3]

  p = - s 2 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y6, y4, 0, y5, y2, -y6 + y3, y3, y5, y1, y6 + y5]

  p' = - s 5 + s 8   p = - s 4 + s 7   p' = - s 4 + s 7


 
723 . Coloring, {2, 5, 7, 9, 11}

R: [7, 8, 7, 6, 3, A, A, C, C, C, 4, 5]
B: [6, 7, 8, 7, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y3, 2 y5, y1, y2, y6, y5, 0, y4, 0, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y7, y7, y8, 0]

  p = - s 4 + s 9


 
724 . Coloring, {2, 5, 7, 9, 12}

R: [7, 8, 7, 6, 3, A, A, C, C, C, 1, 9]
B: [6, 7, 8, 7, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 6 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, y4, 0, 0, y4, y3, y4, y2, y1, 0, y5]

  p = s 4 - s 8   p' = s 5 - s 7   p' = s 4 - s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1 - y5 + 2 y4, y1, y2, 4 y1 - 2 y5, 2 y1 - y5, y3, y4, 0, y5, y6, 0]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
725 . Coloring, {2, 5, 7, 10, 11}

R: [7, 8, 7, 6, 3, A, A, C, B, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 6 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, y4, y6, y7, y5, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y2, 0, 3 y1, 0, 0, -3 y2 - 3 y1 - 3 y6 + 10 y5 - 3 y3 + 4 y4, 6 y4, 3 y6, 3 y5, 3 y4, 3 y3, 3 y5 + 3 y4]

  p = - s 2 + s 7   p' = s 3 - s 8   p' = s 2 - s 7


 
726 . Coloring, {2, 5, 7, 10, 12}

R: [7, 8, 7, 6, 3, A, A, C, B, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y7, 0, 0, y7, y3, y4, y5, y6, y8, y9]

  p = - s 2 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, -2 y6 - 2 y5 + 5 y4 + 5 y2 - 4 y1, y4, y3, 4 y6 + 4 y5 - 6 y4 - 6 y2 + 5 y1 - y3, 0, 5 y6 + 5 y5 - 8 y4 - 8 y2 + 6 y1, y2, y1]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
727 . Coloring, {2, 5, 7, 11, 12}

R: [7, 8, 7, 6, 3, A, A, C, B, C, 4, 9]
B: [6, 7, 8, 7, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 10 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, y2, y3, y4, y7, y5, y6, y8]

  p = s 3 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[y9, y10, y8, 0, y7, y6, y4, y5, 0, y2, y3, y1]


 
728 . Coloring, {2, 5, 8, 9, 10}

R: [7, 8, 7, 6, 3, A, B, B, C, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, -y6 + y5 + 3 y4, y4, y3, y2, 0, -y6 + y5 + 2 y4, y1, y4]

  p' = s 5 - s 8   p = s 5 - s 8   p' = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y9, y8, 0, y7, y6, y5, y4, y3, y2, y1]


 
729 . Coloring, {2, 5, 8, 9, 11}

R: [7, 8, 7, 6, 3, A, B, B, C, C, 4, 5]
B: [6, 7, 8, 7, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y2 - y3 + y4 + y5 + y6 - y7 - y8, y1, y2, y3, y4, y5, 0, y6, y7, y8]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}, {2, 7, 10}}   

See Matrix
 

[9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y6 + 9 y7 - 7 y8 - 7 y9, 7 y1, 7 y2, 0, 0, 7 y3, 7 y4, 7 y5, 7 y6, 7 y7, 7 y8, 7 y9]

  p = - s - s 2 - s 3 + s 8 + s 9 + s 10


 
730 . Coloring, {2, 5, 8, 9, 12}

R: [7, 8, 7, 6, 3, A, B, B, C, C, 1, 9]
B: [6, 7, 8, 7, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y4 + 3 y1, 0, 3 y4, 0, 0, 3 y4, -3 y4 + 3 y1, 3 y4, 3 y2, 3 y3, 3 y1, -3 y4 + 7 y1 - 3 y2 - 3 y3]

  p = s 3 - s 7   p' = s 3 - s 5   p' = s 4 - s 6   p' = - s 5 + s 7   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, y6 - y7, y6, y1, y7, y2, y3, 0, y4, y7, y8]

  p = - s 6 + s 9   p' = - s 6 + s 9


 
731 . Coloring, {2, 5, 8, 10, 11}

R: [7, 8, 7, 6, 3, A, B, B, B, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, -y3 + y1 + y2 - y4 + y5 + y6 + y7 - y8, y1, y2, y4, y5, y6, 0, y7, y8, 0]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, -y2 + y5, y2, y3, y4, y5, 0, y6]

  p = s 5 - s 7   p' = s 5 - s 7


 
732 . Coloring, {2, 5, 8, 10, 12}

R: [7, 8, 7, 6, 3, A, B, B, B, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, y7, y5, 0, 0, y5, y3, y4, 2 y5, y1, y2, 0]

  p' = - s 5 + s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y6, y2, y6, 2 y1, y3, 0, y4, 0, y5]

  p = s 4 - s 7   p' = s 4 - s 7


 
733 . Coloring, {2, 5, 8, 11, 12}

R: [7, 8, 7, 6, 3, A, B, B, B, C, 4, 9]
B: [6, 7, 8, 7, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1 - y2 - y4 - y7 + y5 + y6, 0, y1, y2, y3, y4, y7, y5, y6]

  p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = s 3 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y8, 0, y7, 0, y9]


 
734 . Coloring, {2, 5, 9, 10, 11}

R: [7, 8, 7, 6, 3, A, B, C, C, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y5 + y6 + y7 - y8 - y9, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y4, y5, y6, y7, y8, y3, y9]


 
735 . Coloring, {2, 5, 9, 10, 12}

R: [7, 8, 7, 6, 3, A, B, C, C, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y4, 2 y3, -9 y4 + 7 y3 - 9 y2 + 7 y1 + 7 y8 + 7 y7 - 9 y6 + 7 y5, 0, 0, -9 y4 + 7 y3 - 9 y2 + 7 y1 + 7 y8 + 7 y7 - 9 y6 + 7 y5, 2 y2, 2 y1, 2 y8, 2 y7, 2 y6, 2 y5]

  p = s 5 - s 7 - s 8 + s 10   p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y6, y7, y8, y5, y4, 0, y2, y3, y9]


 
736 . Coloring, {2, 5, 9, 11, 12}

R: [7, 8, 7, 6, 3, A, B, C, C, C, 4, 9]
B: [6, 7, 8, 7, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, y7, y3, y2, y1, y6 - y5 + y4 + y3 + y2 - y1]

  p' = s 7 - s 8   p = s 7 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[7 y8, 7 y6, 7 y7, 0, 7 y3, 7 y4, 7 y5, 7 y2, 0, 9 y8 - 7 y6 + 9 y7 - 7 y3 + 9 y4 - 7 y5 + 9 y2 + 9 y1, 7 y1, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
737 . Coloring, {2, 5, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, B, C, B, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y6, y5, y4, 0, y3, y2, y1, y6 + y5 - y4 + y3 - y2 - y1 - y9 + y8 + y7, y9, y8, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[3 y1, 0, 3 y2, 0, 5 y1 + 5 y2 + 5 y3 - 3 y4 + 5 y5 - 3 y6 + 5 y7 - 3 y8, 3 y3, 3 y4, 3 y5, 0, 3 y6, 3 y7, 3 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
738 . Coloring, {2, 6, 7, 8, 9}

R: [7, 8, 7, 6, A, 3, A, B, C, C, 1, 5]
B: [6, 7, 8, 7, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y6, 0, y4, y3, y2, y3, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2 + y7, y2, y1, 0, y2, y3, y5, y4, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8   p' = - s 6 + s 9


 
739 . Coloring, {2, 6, 7, 8, 10}

R: [7, 8, 7, 6, A, 3, A, B, B, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 4 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y8, 0, 2 y6, y6, y5, y7, 0, y3, y4, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, y3, y2, 0, y3, y2, y1, -2 y3 + 5 y2 - 2 y1 - y4, y1, y2, y4]

  p' = s 5 - s 7   p' = s 4 - s 8   p' = s 3 - s 7   p = s 3 - s 9   p' = s 6 - s 8


 
740 . Coloring, {2, 6, 7, 8, 11}

Ωp(Δ)=0:     p = s + s 2 - 2s 3 + 3s 4 + 8s 5 - 8s 7 - 16s 8

R: [7, 8, 7, 6, A, 3, A, B, B, C, 4, 5]
B: [6, 7, 8, 7, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y9, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 0, 3 y4, 3 y3, 3 y2, 3 y1, -3 y7 - 3 y3 + 5 y2 + 5 y1, -3 y6 - 3 y4 + 5 y2 + 5 y1, -3 y5 + 3 y2 + 3 y1]

  p' = - s 3 + s 9   p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
741 . Coloring, {2, 6, 7, 8, 12}

R: [7, 8, 7, 6, A, 3, A, B, B, C, 1, 9]
B: [6, 7, 8, 7, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y7, 0, 0, y8, y6, y8, y3, y2, y4, y5]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -3 y2 + 10 y1 - 3 y7 - 3 y3 + 10 y4 - 3 y5 - 3 y6, 3 y1 + 3 y4 - 3 y8, 3 y2, 3 y1, 3 y7, 3 y3, 3 y4, 0, 3 y5, 3 y6, 3 y8]

  p = s 4 + s 6 - s 7 - s 9   p' = - s 4 - s 6 + s 7 + s 9


 
742 . Coloring, {2, 6, 7, 9, 10}

R: [7, 8, 7, 6, A, 3, A, C, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y4, y1, y2, 0, y3, y4, y5, y6, 0, y7, 0, y8]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y2, y3, y5, y4, y5, y6, y7]

  p = - s 5 + s 8   p' = s 5 - s 8


 
743 . Coloring, {2, 6, 7, 9, 11}

Ωp(Δ)=0:     p = s + s 2 - 2s 3 - 3s 4 - 4s 5 + 8s 6 + 8s 7 + 16s 8

R: [7, 8, 7, 6, A, 3, A, C, C, C, 4, 5]
B: [6, 7, 8, 7, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y7, 2 y3, y6, y4, y5, y3, 0, y2, 0, y1]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 - y2 + y3 - y4 - y5 - y6 + y7, y1, y2, 0, 0, y3, y4, y5, 2 y2, y6, y7, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
744 . Coloring, {2, 6, 7, 9, 12}

R: [7, 8, 7, 6, A, 3, A, C, C, C, 1, 9]
B: [6, 7, 8, 7, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y4, 0, y2, 0, 0, y4, y3, y4, y5, y6, 0, y1]

  p = - s 5 + s 7   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, -y3 + 2 y5, y2, 2 y3, y3, y4, 2 y1 - 2 y3 - y5, 0, y5, y6, 0]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
745 . Coloring, {2, 6, 7, 10, 11}

R: [7, 8, 7, 6, A, 3, A, C, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[0, y7, y6, y5, y4, y3, y2, y1, 0, y10, y9, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[-y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y2, 2 y1, y3, y4, y5, y6, y7]

  p = s 7 - s 8   p' = - s 7 + s 8


 
746 . Coloring, {2, 6, 7, 10, 12}

R: [7, 8, 7, 6, A, 3, A, C, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 9 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y7, y6, y5, 0, 0, y4, y3, y2, y1, y10, y9, y8]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, 0, y7, y8, y9, y4, y5, y6, 0, y2, y3, y1]


 
747 . Coloring, {2, 6, 7, 11, 12}

R: [7, 8, 7, 6, A, 3, A, C, B, C, 4, 9]
B: [6, 7, 8, 7, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y6 + y5 - y7 + y1 - y2 - y3 - y4 + y9 + y8, y6, y5, 0, y7, y1, y2, y3, 0, y4, y9, y8]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
748 . Coloring, {2, 6, 8, 9, 10}

R: [7, 8, 7, 6, A, 3, B, B, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3 - y8, 0, y3, y8, y4, y5, 0, y6, y7, y8]

  p' = - s 6 + s 9   p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y1 + y5 - y6 - y7 - y4 + y3 + y2, 0, y1, y5, y6, y7, y4, y3, y2]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


 
749 . Coloring, {2, 6, 8, 9, 11}

R: [7, 8, 7, 6, A, 3, B, B, C, C, 4, 5]
B: [6, 7, 8, 7, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -3 y6 + 5 y5 - 3 y2 - 3 y3 - 3 y4 + 5 y1 - 3 y7 + 5 y8, 3 y6, 3 y5, 3 y2, 3 y3, 3 y4, 0, 3 y1, 3 y7, 3 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y2, y1, y4, 0, 0, y3, y5, y6, y7, y8, y9, y10]


 
750 . Coloring, {2, 6, 8, 9, 12}

R: [7, 8, 7, 6, A, 3, B, B, C, C, 1, 9]
B: [6, 7, 8, 7, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 6 y3 - 3 y2 - 3 y4 + 10 y5, 0, 3 y1, 0, 0, 3 y3, 3 y2, 3 y3, -3 y3 + 3 y5, 3 y3, 3 y4, 3 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 11 y1 - 5 y2 + 11 y3 - 10 y5 - 5 y7 + 11 y8 - 5 y4 + 11 y6, 5 y1, 5 y2, 5 y3, 5 y5, 5 y7, 5 y8, 0, 5 y4, 5 y5, 5 y6]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9   p = s 3 - s 6 - s 7 + s 10


 
751 . Coloring, {2, 6, 8, 10, 11}

R: [7, 8, 7, 6, A, 3, B, B, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 4 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y9, y8, y7, y6, y5, y4, y3, 0, y2, y1, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[-2 y1 + 4 y2 + 4 y4 - 4 y3, 0, -y1 + 2 y2 + 2 y4 - 2 y3, 0, 0, y1, -2 y1 + 4 y2 + 4 y4 - 4 y3, y2 + y4 - y3, y2, y4, 0, y3]

  p' = - s 4 + s 7   p' = - s 4 + s 6   p' = - s 4 + s 5   p = s 4 - s 5


 
752 . Coloring, {2, 6, 8, 10, 12}

R: [7, 8, 7, 6, A, 3, B, B, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, y5, y4, 0, 0, y5 - y4, y3, y2, 2 y5 - 2 y4, y5 - y4, y1, 0]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y2, 2 y4, y3, y4, -y2 + y3 + y4, y3 + 2 y4 - y1, 0, y1, 0, y3 + 2 y4]

  p = - s 4 + s 7   p = - s 4 + s 8   p = - s 4 + s 5   p = - s 4 + s 6


 
753 . Coloring, {2, 6, 8, 11, 12}

R: [7, 8, 7, 6, A, 3, B, B, B, C, 4, 9]
B: [6, 7, 8, 7, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y7, y5, y7, y6, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 3 y4, 3 y3, 3 y2, 3 y1, 0, -3 y7 - 3 y6 + 10 y4 - 3 y3 - 3 y2 + 10 y1, 0, -3 y5 + 3 y4 + 3 y1]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9   p = - s 3 - s 5 + s 6 + s 8


 
754 . Coloring, {2, 6, 9, 10, 11}

R: [7, 8, 7, 6, A, 3, B, C, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y2 - y4 + y3, -y5 + y1 + y3, y2 - y1 + y5, y2 + y5 - y4, y2, y1 - y4 + y3, y1, 0, y4, y5, y3]

  p' = - s 2 + s 7   p' = - s 4 + s 9   p = - s + s 6   p' = - s + s 6   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[3 y1 + y7 - y6 - y5 - y4 + y3 + y2, 0, y1, 0, 0, y7, 2 y1, y6, y5, y4, y3, y2]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
755 . Coloring, {2, 6, 9, 10, 12}

R: [7, 8, 7, 6, A, 3, B, C, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 10 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y4 + y2 + y3 + y6 - y1 - y5, y4 + y2 + y3 - y6, y2 + y3 - y6, 0, 0, y4, y1, y2, y3, y4, y5, y6]

  p' = - s 4 + s 7   p' = - s 5 + s 8   p = - s 4 + s 10   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, -y2 - y3 + y5 + y6, -y1 + y5 + y6, y1, y2, y3, -y4 + y5 + y6, 0, y4, y5, y6]

  p = - s 2 + s 3 - s 6 + s 7   p = - s 2 + s 4 - s 6 + s 8   p = - s 2 + s 5 - s 6 + s 9


 
756 . Coloring, {2, 6, 9, 11, 12}

R: [7, 8, 7, 6, A, 3, B, C, C, C, 4, 9]
B: [6, 7, 8, 7, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, 0, -y1 - y2 - y3 + 2 y6 - y5, y1, 0, y2, y3, y4, -2 y4 + y6, y4, y5, y6]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y3, y1, y2, 0, y8, y9, y7, y6, 0, y5, y4, 0]


 
757 . Coloring, {2, 6, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, B, C, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 6 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, y3, y1, y2, 0, y10, y9, y6, y7, y8, y5, y4]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y4, 0, y5, 0, y6, y4 - y5 + y6 - y1, y1, y4 + y6 - y2, 0, y2, y4 + y6 - y3, y3]

  p' = - s 2 + s 3 - s 6 + s 7   p = s 2 - s 3 + s 6 - s 7   p' = - s 2 + s 4 - s 6 + s 8


 
758 . Coloring, {2, 7, 8, 9, 10}

R: [7, 8, 7, 6, A, A, A, B, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y6, y7, 0, 0, y5, y4, y2, y3, 0, -y6 + y7 + y5 + y2 - y3 + y1, y1, y4]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y5, y6, y4, y3, 0, y7, y8]


 
759 . Coloring, {2, 7, 8, 9, 11}

R: [7, 8, 7, 6, A, A, A, B, C, C, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y3, y4, 2 y5, y5, 0, y6, y7, y2]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y3, y2, y1, 0, 0, y8, y9, y7, y6, 0, y5, y4]


 
760 . Coloring, {2, 7, 8, 9, 12}

R: [7, 8, 7, 6, A, A, A, B, C, C, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 18 y1 - 14 y2 + 18 y3 - 14 y7 + 18 y4 - 14 y5 + 18 y6, 5 y1, 5 y2, 5 y3, 9 y1 - 7 y2 + 9 y3 - 7 y7 + 9 y4 - 7 y5 + 9 y6, 5 y7, 5 y4, 0, 0, 5 y5, 5 y6]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8   p = - s 2 + s 5 + s 6 - s 9


 
761 . Coloring, {2, 7, 8, 10, 11}

R: [7, 8, 7, 6, A, A, A, B, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, y4, y5, y4, y3, 0, y7, y6, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y7, 0, 0, y6, y2, y3, y4, 0, y5, y8]


 
762 . Coloring, {2, 7, 8, 10, 12}

R: [7, 8, 7, 6, A, A, A, B, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y5, y4, 0, 0, 0, y3, y2, y1, 2 y3, y7, y6, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 0, y3, y5, y2, y1, y5, -y3 - y2 - y1 + 5 y5 - y4, 0, 0, y5, y4]

  p' = - s 3 + s 7   p' = - s 2 + s 6   p = - s 2 + s 6


 
763 . Coloring, {2, 7, 8, 11, 12}

R: [7, 8, 7, 6, A, A, A, B, B, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y1, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y2, y1, -y2 + y1 + y8 + y7 - y6 + y5 + y4 - y3, 0, y8, y7, y6, y5, 0, 0, y4, y3]

  p = s 6 - s 7 + s 8 - s 9


 
764 . Coloring, {2, 7, 9, 10, 11}

R: [7, 8, 7, 6, A, A, A, C, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y5, y6, y4, y5, y3, 0, y2, 0, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y7, y4, y5, 0, y6, y7]

  p = - s 3 + s 8


 
765 . Coloring, {2, 7, 9, 10, 12}

R: [7, 8, 7, 6, A, A, A, C, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y6, y7, 0, 0, 0, y6, y4, y5, y3, y2, 0, y1]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y3, y4, y2, y5, y7, 0, 0, y6, 2 y2]

  p = - s 5 + s 8


 
766 . Coloring, {2, 7, 9, 11, 12}

R: [7, 8, 7, 6, A, A, A, C, C, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y2, 0, y1, 2 y2, y2, y3, y4, 0, y5]

  p = s 4 - s 6   p' = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y7, y5, y6, 0, y5, y4, y2, y3, 0, 0, y1, 0]

  p = - s 3 + s 8


 
767 . Coloring, {2, 7, 10, 11, 12}

R: [7, 8, 7, 6, A, A, A, C, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y4, 0, y3, y2, y8, y9, y6, y7, y5]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[2 y7, 0, 2 y6, 0, 2 y4, 2 y5, 2 y3, 2 y2, 0, 0, 2 y1, 3 y3]

  p = - s 3 + s 8


 
768 . Coloring, {2, 8, 9, 10, 11}

R: [7, 8, 7, 6, A, A, B, B, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y1 + y7 + y6 - y5 - y4 + y3 - 3 y2, y7, y6, 2 y2, y5, 0, y4, y3, y2]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y6, y7, y4, y5, y3, y9, y8]


 
769 . Coloring, {2, 8, 9, 10, 12}

R: [7, 8, 7, 6, A, A, B, B, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1 + y2 - y3 + y4 + y5 - y6, y1, y2, y7, y3, y4, 0, y5, y7, y6]

  p = s 5 - s 6 + s 7 - s 8   p = - s 5 + s 9


 
770 . Coloring, {2, 8, 9, 11, 12}

R: [7, 8, 7, 6, A, A, B, B, C, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, y2, 2 y1, y1, y7, y6, y5, y4]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[-2 y1 + 5 y6 - 2 y2, 2 y6, 2 y1, 0, 2 y4, 2 y5, 2 y6, 2 y3, 0, 2 y6, -2 y4 - 2 y5 - 2 y3 + 5 y6, 2 y2]

  p = s 4 - s 8   p' = s 4 - s 8   p' = s 6 - s 7 + s 8 - s 9   p' = s 5 - s 9


 
771 . Coloring, {2, 8, 10, 11, 12}

R: [7, 8, 7, 6, A, A, B, B, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, y3, y6, y7, y6, y5, y4, 0]

  p = s 2 - s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[-2 y4 + 2 y2, 0, y6, 0, y5, y4, -2 y4 + 2 y2, y3, 0, y2, 0, y1]

  p' = - s 3 + s 7   p = - s 3 + s 7


 
772 . Coloring, {2, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, A, B, C, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1, 0, y4, 0, y3, y2, y8, y6, y7, y5, y9]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y9, y8, y7, y6, 0, y5, y4, y3]


 
773 . Coloring, {3, 4, 5, 6, 7}

R: [7, 7, 8, 7, 3, 3, A, C, B, C, 1, 5]
B: [6, 8, 7, 6, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, y3, 0, y2, y1, 0, y8, y7, y6]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y7, 0, 3 y6, 0, 3 y5, 3 y3, 3 y4, 3 y2, -3 y6 - 3 y3 - 3 y4 + 5 y2 + 8 y1, -3 y7 - 3 y5 + 8 y2 + 5 y1, 3 y1]

  p' = s 2 - s 8   p = s 2 - s 8


 
774 . Coloring, {3, 4, 5, 6, 8}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, C, 1, 5]
B: [6, 8, 7, 6, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, 0, y6, 0, y5, 0, y4, y3, 0, 0, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y5, 0, 2 y3, 0, y4, y3, y2, y7, y1, 0, y6]

  p = s 6 - s 8


 
775 . Coloring, {3, 4, 5, 6, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, C, 1, 5]
B: [6, 8, 7, 6, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[7 y6, 0, 7 y5, 0, 7 y4, 0, 7 y2, 7 y3, 0, 0, 7 y1, 9 y6 - 7 y5 - 7 y4 + 9 y2 - 7 y3 + 9 y1]

  p = s + s 2 + s 3 - s 5 - s 6 - s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, y5, 2 y4, y6, y7, 0]

  p = - s 2 + s 8


 
776 . Coloring, {3, 4, 5, 6, 10}

R: [7, 7, 8, 7, 3, 3, B, C, B, 2, 1, 5]
B: [6, 8, 7, 6, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[3 y6, 3 y5, 3 y4, 0, 3 y3, 0, 3 y2, 3 y1, 0, 0, -3 y6 - 3 y5 - 3 y2 + 10 y3 + 10 y1, -3 y4 + 3 y3 + 3 y1]

  p' = s 2 + s 4 - s 5 - s 7   p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y7, y6, y6, y5, y4, y3, y2]

  p = s 6 - s 8


 
777 . Coloring, {3, 4, 5, 6, 11}

R: [7, 7, 8, 7, 3, 3, B, C, B, C, 4, 5]
B: [6, 8, 7, 6, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, y1 - y2 + y4 - y3 + y6 - y5, y1, y2, 0, y4, y3, 0, 0, y6, y5]

  p = s + s 2 + s 3 - s 5 - s 6 - s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[3 y7, 3 y6, 0, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y7 - 3 y3 + 8 y2 + 5 y1, -3 y6 - 3 y5 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = - s 2 + s 8   p = - s 2 + s 8


 
778 . Coloring, {3, 4, 5, 6, 12}

R: [7, 7, 8, 7, 3, 3, B, C, B, C, 1, 9]
B: [6, 8, 7, 6, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, 0, y7, y6, y5, 0, y4, y3]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, 0, y2, y1, -y3 + y2 - y1 + y6 + y5 - y4, y7, y6, 0, y5, y4, y7]

  p' = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = s 3 - s 9


 
779 . Coloring, {3, 4, 5, 7, 8}

R: [7, 7, 8, 7, 3, A, A, B, B, C, 1, 5]
B: [6, 8, 7, 6, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, -y1 + y4 + y5, 0, -y2 + y4 + y5, -y3 + y4 + y5, 0, y3, y4, y5]

  p = s - s 2 + s 5 - s 6   p' = - s + s 3 - s 5 + s 7   p' = - s + s 2 - s 5 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, y8, y7, y6, 0, y5, y4, y2, y3, y1, -y7 - y6 - y5 - y4 + 2 y2 + 2 y3 + 2 y1, -y8 + y2 + y3 + y1]

  p = - s 4 + s 9   p' = - s 4 + s 9


 
780 . Coloring, {3, 4, 5, 7, 9}

R: [7, 7, 8, 7, 3, A, A, C, C, C, 1, 5]
B: [6, 8, 7, 6, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y4, 0, y3, 0, y2, 0, -y3 + y4 + y2, y1, 0, y4 + y2 - y1, 0, y4 + y2]

  p' = - s 5 + s 6   p' = s 4 - s 5   p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y7, y7, y8, 0]

  p = - s 4 + s 9


 
781 . Coloring, {3, 4, 5, 7, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[-5 y1 + 11 y2 + 11 y3 - 5 y4 + 11 y5 - 5 y6 - 5 y7 + 11 y8, 5 y1, 5 y2, 0, 5 y3, 0, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 3 y6, 3 y5, 0, 3 y4, 3 y3, 3 y2, 3 y1, 3 y2, -3 y6 - 3 y5 - 3 y4 - 3 y3 + 7 y2 + 10 y1, 3 y2 + 3 y1]

  p' = s 3 - s 8   p' = s 2 - s 7   p = s 2 - s 7


 
782 . Coloring, {3, 4, 5, 7, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, C, 4, 5]
B: [6, 8, 7, 6, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, -y4 + y3 + y2, y3 + y2 - y5, y5, 0, y4, y3, 0, y2, y1, y3 + y2 - y1]

  p' = s 6 - s 7   p' = s 5 - s 7   p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[3 y9, 3 y7, 3 y8, 0, 0, 3 y5, 3 y6, 3 y3, 3 y4, -3 y9 - 3 y7 - 3 y8 - 3 y5 - 3 y6 - 3 y3 + 13 y4 - 3 y1 + 13 y2, 3 y1, 3 y2]

  p = - s 4 - s 5 + s 9 + s 10


 
783 . Coloring, {3, 4, 5, 7, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, C, 1, 9]
B: [6, 8, 7, 6, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y1 - y7 + y6 + y5 + y4 - y3 - y2, 0, 0, 0, y7, y6, y5, y4, y3, y2]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, y9, y10, y8, y6, y7, y5, y4, 0, y3, y1, y2]


 
784 . Coloring, {3, 4, 5, 8, 9}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, C, 1, 5]
B: [6, 8, 7, 6, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y3, 0, y6, y7, 0, y4, y5, y8]

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[0, y10, y9, y8, 0, y7, y6, y5, y4, y3, y2, y1]


 
785 . Coloring, {3, 4, 5, 8, 10}

R: [7, 7, 8, 7, 3, A, B, B, B, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, 2 y1 - 3 y4, 0, 2 y4, 0, y6, y5, 0, y4, y3, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8


 
786 . Coloring, {3, 4, 5, 8, 11}

R: [7, 7, 8, 7, 3, A, B, B, B, C, 4, 5]
B: [6, 8, 7, 6, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y1, y8, 0, y7, y6, 0, y5, y4, y3]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, 0, y9]


 
787 . Coloring, {3, 4, 5, 8, 12}

R: [7, 7, 8, 7, 3, A, B, B, B, C, 1, 9]
B: [6, 8, 7, 6, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, 0, y6, y5, y4, y3, y2, y3 + y5]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y7, y6, 0, y8, 0, y9]


 
788 . Coloring, {3, 4, 5, 9, 10}

R: [7, 7, 8, 7, 3, A, B, C, C, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[3 y3, 3 y2, 3 y1, 0, 3 y1 - 3 y7 + 3 y4, 0, -3 y3 - 3 y2 + 10 y1 - 3 y6 - 3 y5 + 10 y4, 3 y7, 0, 3 y6, 3 y5, 3 y4]

  p' = - s 3 - s 5 + s 6 + s 8   p = s 3 + s 5 - s 6 - s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y9, y8, 0, y7, y5, y6, y3, y2, y4, y1]


 
789 . Coloring, {3, 4, 5, 9, 11}

R: [7, 7, 8, 7, 3, A, B, C, C, C, 4, 5]
B: [6, 8, 7, 6, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 0, 9 y1 - 7 y4 + 9 y2 - 7 y3 - 7 y5 + 9 y6 - 7 y7, 7 y1, 7 y4, 0, 7 y2, 7 y3, 0, 7 y5, 7 y6, 7 y7]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, y9, 0]


 
790 . Coloring, {3, 4, 5, 9, 12}

R: [7, 7, 8, 7, 3, A, B, C, C, C, 1, 9]
B: [6, 8, 7, 6, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[7 y1, 0, 7 y4, 0, 0, 0, 7 y2, 7 y6, 7 y5, 7 y4, 7 y3, 9 y1 - 14 y4 + 9 y2 - 7 y6 - 7 y5 + 9 y3]

  p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y4, y5, y3, y1, y2, y6, y9, 0, y8, y7, 0]


 
791 . Coloring, {3, 4, 5, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 11 y1 - 5 y2 + 11 y3 - 5 y8 + 11 y4 - 5 y5 - 5 y6 + 11 y7, 5 y1, 5 y2, 5 y3, 0, 5 y8, 5 y4, 0, 5 y5, 5 y6, 5 y7]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y2, 0, y1, 0, 0, y3, y4, y5, y8, y6, y7, y9]


 
792 . Coloring, {3, 4, 5, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y5 + y3, y5, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y6, y3, y4, y5, 0, y7, y8, y9]


 
793 . Coloring, {3, 4, 5, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, C, 4, 9]
B: [6, 8, 7, 6, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, 0, y7, y2, y3, y4, y5, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y9, y8, y7, 0, y6, y4, y5, y3, 0, y2, y1, -y9 + y8 - y7 + y6 + y4 + y5 - y3 - y2 + y1]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10


 
794 . Coloring, {3, 4, 6, 7, 8}

R: [7, 7, 8, 7, A, 3, A, B, B, C, 1, 5]
B: [6, 8, 7, 6, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y7, 0, y3, y4, 0, y5, y6, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   10

See Matrix
 

[0, y9, y10, y8, 0, y7, y6, y5, y4, y3, y1, y2]


 
795 . Coloring, {3, 4, 6, 7, 9}

R: [7, 7, 8, 7, A, 3, A, C, C, C, 1, 5]
B: [6, 8, 7, 6, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, y3, 0, y2, 0, y3 + 2 y1, y1, 0, y5, 0, y4]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, y4, y7, y5, 2 y2, y6, y8, 0]

  p = - s 3 + s 9


 
796 . Coloring, {3, 4, 6, 7, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6 + y4, y1, y6, 0, y2, 0, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y4, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 7 + s 9


 
797 . Coloring, {3, 4, 6, 7, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, C, 4, 5]
B: [6, 8, 7, 6, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y5, y5 + y3, y1, 0, y2, y3, 0, y4, y5, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[2 y2, 11 y2 - 2 y1 - 2 y4 + 11 y5 + 11 y6 + 11 y3 - 2 y7 - 39 y8, 2 y1, 0, 0, 2 y4, 2 y5, 2 y6, 3 y2 + 3 y5 + 3 y6 + 3 y3 - 11 y8, 2 y3, 2 y7, 2 y8]

  p = - s 3 + s 9   p' = - s 3 + s 9


 
798 . Coloring, {3, 4, 6, 7, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, C, 1, 9]
B: [6, 8, 7, 6, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 + y2 - y4 - y3 - y6 + y5 + y7, 0, y1, 0, 0, 0, y2, y4, y3, y6, y5, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y8, y7, y6, y5, y4, y3, y2, 0, y1, y10, y9]


 
799 . Coloring, {3, 4, 6, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, C, 1, 5]
B: [6, 8, 7, 6, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {1, 7, 11}}    order:   3

See Matrix
 

[y1, 0, y2, 0, y3, 0, 5 y3 - y4 + 4 y1 + 4 y2 - 6 y5, y4, 0, -2 y1 - 2 y2 - 4 y3 + 5 y5, y5, 6 y3 + 5 y1 + 5 y2 - 8 y5]

  p' = - s 4 + s 7   p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y4, y10, y6, y7, y8, y9, y5]


 
800 . Coloring, {3, 4, 6, 8, 10}

R: [7, 7, 8, 7, A, 3, B, B, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 2 y4 - y5, 0, 4 y4 - 2 y5, 0, y3, y4, 0, y5, y6, 0 ]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, 2 y4, y4, y5, y3, y2, 0, y1]

  p' = s 4 - s 6   p' = s 5 - s 7   p = s 4 - s 8


 
801 . Coloring, {3, 4, 6, 8, 11}

R: [7, 7, 8, 7, A, 3, B, B, B, C, 4, 5]
B: [6, 8, 7, 6, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -2 y1 + 9 y2 + 9 y3 - 11 y5 - 2 y4, 2 y1, -2 y2 - 2 y3 + 2 y5 + 2 y4, 0, 2 y2, 2 y3, 0, 7 y2 + 7 y3 - 9 y5 - 2 y4, 2 y4, 2 y5]

  p' = s 3 - s 6   p' = s 4 - s 7   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y1, y2, 0, 0, 2 y3, y3, y7, y4, y5, 0, y6]

  p = - s 6 + s 8   p' = - s 6 + s 8


 
802 . Coloring, {3, 4, 6, 8, 12}

R: [7, 7, 8, 7, A, 3, B, B, B, C, 1, 9]
B: [6, 8, 7, 6, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, 0, y2, -y4 + y6, y3, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, 0, y9]


 
803 . Coloring, {3, 4, 6, 9, 10}

R: [7, 7, 8, 7, A, 3, B, C, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   9

See Matrix
 

[y3, y2, y1, 0, y9, 0, y8, y7, 0, y6, y5, y4]

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
804 . Coloring, {3, 4, 6, 9, 11}

R: [7, 7, 8, 7, A, 3, B, C, C, C, 4, 5]
B: [6, 8, 7, 6, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 2 y1, 4 y1 + 4 y5 + 3 y2 + 3 y3 - 7 y4, 2 y5, 0, 10 y1 + 10 y5 + 4 y2 + 4 y3 - 16 y4, 2 y2, 0, 2 y3, 2 y4, 16 y1 + 16 y5 + 7 y2 + 7 y3 - 27 y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y8, y7, y6, 0, 0, y4, y5, y3, 2 y6, y2, y1, 0]

  p = - s 3 + s 9


 
805 . Coloring, {3, 4, 6, 9, 12}

R: [7, 7, 8, 7, A, 3, B, C, C, C, 1, 9]
B: [6, 8, 7, 6, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[7 y1, 0, 7 y3, 0, 0, 0, 7 y6, 7 y5, 7 y4, 7 y3, 7 y2, 9 y1 - 14 y3 + 9 y6 - 7 y5 - 7 y4 + 9 y2]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y2, y1, y8, y7, y6, y5, y4, 0, y3, y9, 0]


 
806 . Coloring, {3, 4, 6, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y8, y9, y7, y6, 0, y5, y4, 0, y3, y1, y2]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y5, y7, y3, y2, y4, y7, y6]

  p = s 6 - s 8   p' = s 6 - s 8


 
807 . Coloring, {3, 4, 6, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y4 + y7, y4, 0, 0, 0, y2, y7, y3, y4, y5, y6]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y5, y4, y2, y3, y1, y9, 0, y8, y6, y7]


 
808 . Coloring, {3, 4, 6, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, C, 4, 9]
B: [6, 8, 7, 6, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y6, y2, 0, 0, y1, y3, y7, y6, y5, y4]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y2 + y1 - y3 - y4 - y5 + y6 + y9 - y7 + y8, y1, 0, y3, y4, y5, y6, 0, y9, y7, y8]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
809 . Coloring, {3, 4, 7, 8, 9}

R: [7, 7, 8, 7, A, A, A, B, C, C, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, 0, y3, y4, 0, y6, y5, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y3, y2, 0, y6, y7, y5, y4, 0, y8, y9]


 
810 . Coloring, {3, 4, 7, 8, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 2 y5, 0, y6, y5, 0, y4, y3, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 7 y4, 7 y3, 0, 7 y2, -7 y4 - 7 y3 - 7 y2 + 9 y1 + 9 y7 - 7 y6 + 9 y5, 7 y1, 7 y7, 0, 7 y6, 7 y5]

  p = s 2 + s 3 - s 7 - s 8


 
811 . Coloring, {3, 4, 7, 8, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, 0, y3, y4, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[9 y1 - 7 y2 - 7 y6 - 7 y7 + 9 y8 + 9 y5 - 7 y4 + 9 y3, 7 y1, 7 y2, 0, 0, 7 y6, 7 y7, 7 y8, 7 y5, 0, 7 y4, 7 y3]

  p = - s 3 - s 4 + s 8 + s 9


 
812 . Coloring, {3, 4, 7, 8, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, 0, 0, y1, y7, y6, y5, y4, y3]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y5, y4, y3, y2, y1, y9, y8, 0, 0, y7, y6]


 
813 . Coloring, {3, 4, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y2, y1, 0, 0, y5, 0, y6, y2, 0, y3, 0, y4]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, 0, y7, 2 y5]

  p = - s 3 + s 8


 
814 . Coloring, {3, 4, 7, 9, 11}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y4, y5, 0, y3, y4, 0, y2, 0, y1]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y3, y2, 0, 0, y5, y6, y7, y3, 0, y4, 0]

  p = - s 3 + s 8


 
815 . Coloring, {3, 4, 7, 9, 12}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y3, 0, 0, 0, 0, 0, y4, y3, y2, y1, 0, y5]

  p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y5, y1, y2, y5, y3, y4, y6, 0, 0, y7, 0]

  p = - s 3 + s 8


 
816 . Coloring, {3, 4, 7, 10, 11}

R: [7, 7, 8, 7, A, A, A, C, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, 0, y6, y1, 0, y2, y3, 0, y4, y3, y6 - y3]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[5 y5, 0, 5 y4, 0, 0, 5 y3, 5 y2, -5 y5 - 5 y4 - 5 y3 - 5 y2 + 11 y1 - 5 y7 + 11 y6, 5 y1, 0, 5 y7, 5 y6]

  p = - s 2 - s 3 + s 7 + s 8


 
817 . Coloring, {3, 4, 7, 10, 12}

R: [7, 7, 8, 7, A, A, A, C, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, y4, y5, y8, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y5, y6, y1, y2, y3, y4, 0, 0, y7, 3 y4]

  p = - s 3 + s 8


 
818 . Coloring, {3, 4, 7, 11, 12}

R: [7, 7, 8, 7, A, A, A, C, B, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, y3, y4, y6, y5, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, y3, y4, y5, 2 y3 - 3 y7, 0, 0, y6, y7]

  p' = s 3 - s 8   p = s 3 - s 8


 
819 . Coloring, {3, 4, 8, 9, 10}

R: [7, 7, 8, 7, A, A, B, B, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, y3, 0, 0, y2, 0, y1, y7, 0, y6, y5, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3, y2, 0, y1, y8, y9, y7, y5, y6, y4]


 
820 . Coloring, {3, 4, 8, 9, 11}

R: [7, 7, 8, 7, A, A, B, B, C, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 16 y1 + 16 y2 - 27 y3 + 7 y4, 4 y1 + 4 y2 + 3 y4 - 7 y3, 0, 2 y1, 2 y2, 0, 10 y1 + 10 y2 - 16 y3 + 4 y4, 2 y4, 2 y3]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[y4, y5, y6, 0, 0, y1, y2, y3, y7, y8, y9, y10]


 
821 . Coloring, {3, 4, 8, 9, 12}

R: [7, 7, 8, 7, A, A, B, B, C, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-7 y1 + 11 y4 + 9 y3 - 7 y2 + 9 y5, 0, 0, 0, 0, 0, 7 y1, 7 y4, 7 y3, 14 y4, 7 y2, 7 y5]

  p' = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 4 + s 5 - s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y2, y1, y6, y7, y5, y3, y4, 0, y9, y10, y8]


 
822 . Coloring, {3, 4, 8, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, 0, y3, y4, 0, y5, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, 0, y5, 0, 0, y4, y3, y2, y1, y7, 0, y6]

  p = s 6 - s 8


 
823 . Coloring, {3, 4, 8, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y3, y4, 0, 0, 0, 0, y1, y2, 2 y2, 2 y2, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y7, 2 y3, y6, y5, y4, y3, 0, y2, 0, y1]

  p = s 3 - s 8


 
824 . Coloring, {3, 4, 8, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y6, 0, 0, y5, y4, y3, 2 y4, y2, y1]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, 0, y9]


 
825 . Coloring, {3, 4, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y3, 0, y1, y2, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, 0, y2, y3, y1, y8, y9, y7, y6]


 
826 . Coloring, {3, 4, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, 5 y1 + 5 y2 + 5 y3 - 11 y4 + 5 y5 - 11 y6, y4, 10 y1 + 10 y2 + 10 y3 - 22 y4 + 10 y5 - 22 y6, y5, y6]

  p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
827 . Coloring, {3, 4, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, 7 y1, 0, 0, 7 y3, 7 y4, 7 y5, 14 y4, 7 y2, 9 y1 + 9 y3 - 21 y4 - 7 y5 + 9 y2]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, y6, 0, y7, y8, y9, y5, 0, y3, y4, 0]


 
828 . Coloring, {3, 4, 10, 11, 12}

R: [7, 7, 8, 7, A, A, B, C, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y2, 0, y1, 0, 0, y6, y5, y4, 2 y5, y3, y2]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y9, 0, y8, 0, y6, y7, y5, y4, 0, y3, y1, y2]


 
829 . Coloring, {3, 5, 6, 7, 8}

R: [7, 7, 8, 6, 3, 3, A, B, B, C, 1, 5]
B: [6, 8, 7, 7, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y8, 0, y7, 0, y6, y4, y5, y3, 0, y2, y1, y8 - y7 + y6 + y4 - y5 + y3 + y2 - y1]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 2 y3, 0, 2 y5, 0, -2 y3 + 5 y5 - 2 y4, 2 y5, 2 y2, 5 y5 - 2 y2 - 2 y1, 2 y1, 2 y5, 2 y4]

  p' = s 7 - s 8   p = s 5 - s 9   p' = s 6 - s 8   p' = s 5 - s 8


 
830 . Coloring, {3, 5, 6, 7, 9}

R: [7, 7, 8, 6, 3, 3, A, C, C, C, 1, 5]
B: [6, 8, 7, 7, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y3, 0, y6, 0, y5, y3, y4, y2, 0, y1, 0, -y6 + y5 + 3 y3 - y4 + y2 + y1]

  p = - s 4 + s 8   p = - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, y5, 2 y3, y7, y6, 0]

  p = s 5 - s 8


 
831 . Coloring, {3, 5, 6, 7, 10}

R: [7, 7, 8, 6, 3, 3, A, C, B, 2, 1, 5]
B: [6, 8, 7, 7, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 5 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[2 y8, 2 y6, 2 y7, 0, 2 y5, 7 y8 + 7 y6 - 9 y7 - 9 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y4, 2 y3, 0, 2 y2, 7 y8 + 7 y6 - 9 y7 - 9 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y1 ]

  p = s 3 + s 4 + s 5 - s 7 - s 8 - s 9   p = s 3 - s 6 - s 7 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, -y1 + y2 - y4 + 2 y5, 0, y2, y1, y2, y2 - y3 + y5, y3, y4, y5]

  p = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 3 + s 6


 
832 . Coloring, {3, 5, 6, 7, 11}

R: [7, 7, 8, 6, 3, 3, A, C, B, C, 4, 5]
B: [6, 8, 7, 7, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y1, y2, y3, y4, 2 y6, y5, 0, 2 y2 - 2 y6, y6, y7]

  p' = s 4 - s 8   p = s 4 - s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y5 - 3 y4 + 8 y3 - 3 y2 + 5 y1, 3 y7, 0, 0, 0, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2, -3 y7 - 3 y6 + 5 y3 + 8 y1, 3 y1]

  p' = s 2 - s 8   p = s 2 - s 8


 
833 . Coloring, {3, 5, 6, 7, 12}

R: [7, 7, 8, 6, 3, 3, A, C, B, C, 1, 9]
B: [6, 8, 7, 7, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y8 - y7 + y6 - y5 - y4 - y3 + y2 + y1, 0, y8, 0, 0, y7, y6, y5, y4, y3, y2, y1]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y8, y4, y5, 0, y6, y7, y8]

  p = - s 6 + s 9


 
834 . Coloring, {3, 5, 6, 8, 9}

R: [7, 7, 8, 6, 3, 3, B, B, C, C, 1, 5]
B: [6, 8, 7, 7, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, 0, y1, 0, y5, y3, y4, y7, 0, 0, y6, 3 y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y9, 0, y8, 0, y7, y5, y6, y4, y2, y3, y1]


 
835 . Coloring, {3, 5, 6, 8, 10}

R: [7, 7, 8, 6, 3, 3, B, B, B, 2, 1, 5]
B: [6, 8, 7, 7, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y3, 2 y1, y2, 0, 2 y1, y1, y6, y5, 0, 0, y4, 0]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y2, 0, y2, y1, y2, y3, y4, 0, y5]

  p = s 4 - s 6   p' = - s 4 + s 6


 
836 . Coloring, {3, 5, 6, 8, 11}

R: [7, 7, 8, 6, 3, 3, B, B, B, C, 4, 5]
B: [6, 8, 7, 7, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y7, y6, y5, y4, y3, y2, 0, 0, y1, y3]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y3, y1, 0, 0, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8


 
837 . Coloring, {3, 5, 6, 8, 12}

R: [7, 7, 8, 6, 3, 3, B, B, B, C, 1, 9]
B: [6, 8, 7, 7, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y5, 0, y4, 0, 0, y3, y2, y1, 2 y4 - 2 y3, 0, y6, 2 y3]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, 2 y5, y2, y5, y3, y4, 0, y6, 0, y7]

  p = - s 3 + s 8


 
838 . Coloring, {3, 5, 6, 9, 10}

R: [7, 7, 8, 6, 3, 3, B, C, C, 2, 1, 5]
B: [6, 8, 7, 7, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[y4, 2 y1, y3, 0, y2, y1, -y4 + 2 y3 - 2 y1 - y6 + 2 y5, y3 - y2 - y1 + y5, 0, 0, y6, y5]

  p = - s 2 - s 3 + 2s 6 + s 8 - s 9   p' = - s 3 - s 5 + s 6 + s 8   p = - s 2 + s 5 + s 6 - s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 2 + s 8


 
839 . Coloring, {3, 5, 6, 9, 11}

R: [7, 7, 8, 6, 3, 3, B, C, C, C, 4, 5]
B: [6, 8, 7, 7, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y3, y4, y2, y5, y6, 0, 0, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, y5, y4, y5, y6, y7, 0]

  p = - s 2 + s 8


 
840 . Coloring, {3, 5, 6, 9, 12}

R: [7, 7, 8, 6, 3, 3, B, C, C, C, 1, 9]
B: [6, 8, 7, 7, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y5, 0, y1, 0, 0, 5 y5 - y1 - y4 - y3 - y2, y5, y4, y3, 0, y5, y2]

  p' = - s 5 + s 7   p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y3, y3, y6, y4, 0, y5, y7, 0]

  p = - s 2 + s 8


 
841 . Coloring, {3, 5, 6, 10, 11}

R: [7, 7, 8, 6, 3, 3, B, C, B, 2, 4, 5]
B: [6, 8, 7, 7, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y3, y1, y2, y4, y5, y6, y7, 0, 0, y8, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 2 y3, y3, y4, y6, y5, y7]

  p = - s 6 + s 8


 
842 . Coloring, {3, 5, 6, 10, 12}

R: [7, 7, 8, 6, 3, 3, B, C, B, 2, 1, 9]
B: [6, 8, 7, 7, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y5, y6, 0, 0, y5, y4, y2, y3, 0, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y6, y3, 0, y4, y5, y7]

  p = - s 5 + s 8


 
843 . Coloring, {3, 5, 6, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, C, B, C, 4, 9]
B: [6, 8, 7, 7, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y8, y4, 0, y1, y2, y3, y5, 0, y6, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, 0, y2, y3, 2 y7, y4, 0, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


 
844 . Coloring, {3, 5, 7, 8, 9}

R: [7, 7, 8, 6, 3, A, A, B, C, C, 1, 5]
B: [6, 8, 7, 7, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[-y1 + y5 + y6, 0, y5 + y6 - y2 - y4, 0, y1, y2, y4, -y3 + y5 + y6, 0, y3, y5, y6]

  p' = - s 2 + s 4 - s 6 + s 8   p' = - s 2 + s 3 - s 6 + s 7   p = s 2 - s 3 + s 6 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1 + y6, y1, y2, 0, y6, y3, y4, y5, y6, y7, y8]

  p = - s 6 + s 9   p' = - s 6 + s 9


 
845 . Coloring, {3, 5, 7, 8, 10}

R: [7, 7, 8, 6, 3, A, A, B, B, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 5 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y5, y6, 0, 2 y2, y2, y3, y4, 0, y7, y8, 0]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, y1, -y1 - y5 + 3 y4 - y2 + 2 y3, 0, y4, y5, y4, y3, y4, y2, 2 y4 + y3]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p' = - s 3 + s 6   p = s 3 - s 6


 
846 . Coloring, {3, 5, 7, 8, 11}

R: [7, 7, 8, 6, 3, A, A, B, B, C, 4, 5]
B: [6, 8, 7, 7, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3 + y2 - y5 - y4, -y6 + y3 + y2, y6, y5, y4, y3, 0, y2, y1, y3 + y2 - y1]

  p' = s 2 - s 3 + s 6 - s 7   p' = - s 3 + s 4 - s 7 + s 8   p = s 2 - s 4 + s 6 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[y3, y4, y5, 0, 0, y6, y7, y8, y4 - y8 - y1 + y2, y1, -y3 - y5 - y6 - y7 + 2 y4 + 2 y2, y2]

  p' = - s 4 + s 9   p = - s 4 + s 9


 
847 . Coloring, {3, 5, 7, 8, 12}

R: [7, 7, 8, 6, 3, A, A, B, B, C, 1, 9]
B: [6, 8, 7, 7, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[2 y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y1, y2, y3, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, y1 + y2 - y3 + y4 + y5 - y6 - y7 + y9 - y8, y1, y2, y3, y4, y5, y6, 0, y7, y9, y8]

  p = s 3 + s 4 + s 5 - s 8 - s 9 - s 10


 
848 . Coloring, {3, 5, 7, 9, 10}

R: [7, 7, 8, 6, 3, A, A, C, C, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[6 y2, 10 y1 - 9 y2 - 3 y4 + 10 y3 - 3 y5, 3 y1 + 3 y3 - 3 y6, 0, 3 y1, 3 y2, 3 y4, 3 y3, 0, 3 y5, 0, 3 y6]

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7   p = s 2 - s 5 - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -y4 + y6, y1, 0, y4, y2, y4, y3, y4, y5, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7   p' = - s 5 + s 8


 
849 . Coloring, {3, 5, 7, 9, 11}

R: [7, 7, 8, 6, 3, A, A, C, C, C, 4, 5]
B: [6, 8, 7, 7, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, -y2 + y3 + y4 - y1, y2, -y2 + y3 + y4, y1, y2, y3, 0, y4, 0, y3 + y4]

  p = - s 4 + s 7   p = - s 4 + s 8   p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y7, y4, y5, 2 y6, y6, y8, 0]

  p = - s 4 + s 9


 
850 . Coloring, {3, 5, 7, 9, 12}

R: [7, 7, 8, 6, 3, A, A, C, C, C, 1, 9]
B: [6, 8, 7, 7, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, y1, 0, 0, y1, 2 y2 + 2 y3 - 2 y4, y2 + y3 - y4, y2, y3, 0, y4]

  p' = s 6 - s 7   p' = s 4 - s 7   p' = s 5 - s 7   p = s 4 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y7, y6, y5, 2 y4, y4, y3, y2, 0, 2 y6 - y4, y1, 0]

  p' = s 5 - s 8   p = s 5 - s 8


 
851 . Coloring, {3, 5, 7, 10, 11}

R: [7, 7, 8, 6, 3, A, A, C, B, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 11 y6 - 5 y7 + 11 y8 - 5 y9 - 5 y5 + 11 y3 - 5 y4 - 5 y1 + 11 y2, 5 y6, 5 y7, 5 y8, 5 y9, 5 y5, 5 y3, 0, 5 y4, 5 y1, 5 y2]

  p = - s 4 - s 5 - s 6 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y4, 0, 3 y3, 0, 0, 3 y2, 3 y1, -3 y4 - 3 y3 - 3 y2 - 3 y1 + 10 y5 - 3 y6, 3 y4 + 3 y3 + 3 y2 + 3 y1 - 7 y5 + 3 y6, -3 y4 - 3 y3 - 3 y2 - 3 y1 + 10 y5 - 3 y6, 3 y6, 3 y5]

  p' = s 3 - s 8   p' = s 2 - s 7   p = s 2 - s 7


 
852 . Coloring, {3, 5, 7, 10, 12}

R: [7, 7, 8, 6, 3, A, A, C, B, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, 0, y3, y4, y5, y7, y8, y9, y6]

  p = - s 7 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y3, y4, -2 y3 - 2 y4 + 5 y2 + 5 y1 - 4 y5, y2, 4 y3 + 4 y4 - 7 y2 - 6 y1 + 5 y5, y2, 0, 5 y3 + 5 y4 - 8 y2 - 8 y1 + 6 y5, y1, y5]

  p' = - s 3 + s 6   p = - s 3 + s 9   p' = - s 4 + s 7   p = - s 3 + s 6


 
853 . Coloring, {3, 5, 7, 11, 12}

R: [7, 7, 8, 6, 3, A, A, C, B, C, 4, 9]
B: [6, 8, 7, 7, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 10 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, y5, 2 y7, y4, y3, y2, y1, -3 y7 + y6 - y5 + y4 + y3 + y2 - y1]

  p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = s 3 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y10, y8, y9]


 
854 . Coloring, {3, 5, 8, 9, 10}

R: [7, 7, 8, 6, 3, A, B, B, C, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, y5, y6, 0, -y5 + y6 + 3 y4, y4, y3, y2, 0, -y5 + y6 + 2 y4, y1, y4]

  p' = s 5 - s 8   p' = s 6 - s 9   p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, 0, y8, y7, y8, y4, y3, y1, y2]

  p = s 3 - s 9


 
855 . Coloring, {3, 5, 8, 9, 11}

R: [7, 7, 8, 6, 3, A, B, B, C, C, 4, 5]
B: [6, 8, 7, 7, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y5, y6, y5 - y6 + y1 - y2 - y3 - y4 + y8 + y7, y1, y2, y3, 0, y4, y8, y7]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[y10, y8, y9, 0, 0, y7, y6, y4, y5, y3, y1, y2]


 
856 . Coloring, {3, 5, 8, 9, 12}

R: [7, 7, 8, 6, 3, A, B, B, C, C, 1, 9]
B: [6, 8, 7, 7, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 9 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 + 3 y3, 0, 3 y1, 0, 0, 3 y1, 3 y3 - 3 y2, 3 y2, -3 y1 + 7 y3 - 3 y2 - 3 y4, 3 y2, 3 y3, 3 y4]

  p' = - s 3 + s 7   p = - s 3 + s 9   p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y2 - y7, y2, y3, y7, y4, y5, 0, y6, y7, y8]

  p = s 4 - s 9   p' = - s 4 + s 9


 
857 . Coloring, {3, 5, 8, 10, 11}

Ωp(Δ)=0:     p = s - s 2 - s 4 + 4s 5 - 8s 7 + 16s 8

R: [7, 7, 8, 6, 3, A, B, B, B, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y6 + y5 + y7 - y8, y1, y2, y3, y4, y6, y5, 0, y7, y8, 0]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y5, 0, y1, 0, 0, y7, y6, y5, y4, y3, 0, y2]

  p = - s 6 + s 8


 
858 . Coloring, {3, 5, 8, 10, 12}

R: [7, 7, 8, 6, 3, A, B, B, B, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, y3, 0, 0, y3, y4, y6, 2 y3, y6, y5, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, y2, y4, y3, y4, 0, y5, 0, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
859 . Coloring, {3, 5, 8, 11, 12}

R: [7, 7, 8, 6, 3, A, B, B, B, C, 4, 9]
B: [6, 8, 7, 7, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 - y3 - y4 - y5 + y6 + y7, y1, 0, y2, -2 y1 + 2 y2 - 2 y3 - 2 y4 - 2 y5 + 2 y6 + 2 y7, y3, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y6, y7, y5, 0, y8, 0, y9]


 
860 . Coloring, {3, 5, 9, 10, 11}

R: [7, 7, 8, 6, 3, A, B, C, C, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y1, 3 y4 - 3 y6 + 3 y2, 0, -3 y5 - 3 y1 + 5 y4 + 5 y2, -3 y3 - 3 y7 + 5 y4 + 5 y2, 3 y2]

  p' = s + s 3 - s 7 - s 9   p = - 2s - s 2 - 3s 3 - 2s 5 + 2s 6 + 3s 8 + s 9 + 2s 10   p = s + s 2 + 2s 3 + 2s 5 - 2s 6 + s 7 - 3s 8 - 2s 10

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y9, 0, 0, y8, y7, y4, y5, y6, y3, y2]


 
861 . Coloring, {3, 5, 9, 10, 12}

R: [7, 7, 8, 6, 3, A, B, C, C, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 + 7 y3 + 4 y4 + 3 y6 - 3 y2 - 3 y5, 3 y1, 3 y3 + 3 y4 - 3 y6, 0, 0, 3 y3 + 3 y4 - 3 y6, 3 y2, 3 y4, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8   p = - s 4 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, -y6 + y4, y2, y1, y6, y7, y6, 0, y5, y4, y3]

  p' = s 5 - s 8   p = s 5 - s 8


 
862 . Coloring, {3, 5, 9, 11, 12}

R: [7, 7, 8, 6, 3, A, B, C, C, C, 4, 9]
B: [6, 8, 7, 7, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y6 + y5 + y4 - y3 - y2 + y1, y6, 0, y5, -2 y6 + 2 y5 + 2 y4 - 2 y3 - 2 y2 + 2 y1, y4, y3, y2, 2 y4, y1]

  p' = s 7 - s 8   p' = s 6 - s 8   p = s 6 - s 9

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y6, y5, y4, y7, 0, y9, y8, 0]


 
863 . Coloring, {3, 5, 10, 11, 12}

Ωp(Δ)=0:     p = s 4 - 4s 5 - 8s 6 + 8s 7 - 16s 8

R: [7, 7, 8, 6, 3, A, B, C, B, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y2, y1, y2 + y1 + y3 - y4 - y8 - y7 - y9 + y6 + y5, 0, y3, y4, y8, y7, y9, y6, y5]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y4, 0, y3, 0, y1, y2, y6, y5, 0, y9, y7, y8]


 
864 . Coloring, {3, 6, 7, 8, 9}

R: [7, 7, 8, 6, A, 3, A, B, C, C, 1, 5]
B: [6, 8, 7, 7, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y2, 0, y1, 0, y8, y6, y7, y5, 0, y4, y3, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y3, y2, 0, y3, y9, y4, y5, y6, y7, y8]

  p = - s 7 + s 10


 
865 . Coloring, {3, 6, 7, 8, 10}

R: [7, 7, 8, 6, A, 3, A, B, B, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 9 y6 + 9 y5 - 7 y1 - 7 y2 + 9 y3 - 7 y4, 5 y6, 0, 9 y6 + 9 y5 - 7 y1 - 7 y2 + 9 y3 - 7 y4, 5 y5, 9 y6 + 9 y5 - 7 y1 - 7 y2 + 9 y3 - 7 y4, 5 y1, 5 y2, 5 y3, 5 y4]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8   p = - s 3 + s 9


 
866 . Coloring, {3, 6, 7, 8, 11}

R: [7, 7, 8, 6, A, 3, A, B, B, C, 4, 5]
B: [6, 8, 7, 7, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 0, 7 y8, 7 y7, 7 y6, 7 y5, 7 y4, 7 y3, 0, 7 y2, 7 y1, 9 y8 + 9 y7 - 7 y6 + 9 y5 - 7 y4 + 9 y3 - 7 y2 + 9 y1]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   10

See Matrix
 

[y3, y1, y2, 0, 0, y3 - y1 - y2 + y9 + y7 + y8 + y6 - y4 - y5, y9, y7, y8, y6, y4, y5]

  p = s 9 - s 10


 
867 . Coloring, {3, 6, 7, 8, 12}

R: [7, 7, 8, 6, A, 3, A, B, B, C, 1, 9]
B: [6, 8, 7, 7, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y9, 0, 0, y8, y7, y6, y5, y4, y3, y2]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y2, y7, y3, y4, y5, y6, 0, y9, y8, y10]


 
868 . Coloring, {3, 6, 7, 9, 10}

R: [7, 7, 8, 6, A, 3, A, C, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y4, y1, y2, 0, y3, y4, y5, y6, 0, y7, 0, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y7, y2, 0, y7, y1, y7, y5, y6, y4, y3]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
869 . Coloring, {3, 6, 7, 9, 11}

R: [7, 7, 8, 6, A, 3, A, C, C, C, 4, 5]
B: [6, 8, 7, 7, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, y4, y5, y6, y3, 0, y2, 0, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 - y2 + y3 - y4 - y5 - y6 + y7, y1, y2, 0, 0, y3, y4, y5, 2 y2, y6, y7, 0]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


 
870 . Coloring, {3, 6, 7, 9, 12}

R: [7, 7, 8, 6, A, 3, A, C, C, C, 1, 9]
B: [6, 8, 7, 7, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[-4 y3 + 2 y2, 0, y1, 0, 0, -2 y3 + y2, 2 y1, y3, y4, y2, 0, y5]

  p = s 4 - s 6   p' = s 4 - s 6   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, -y3 + 2 y6, y1, 2 y3, y3, y4, y7, 0, y6, y5, 0]

  p' = s 5 - s 8   p = s 5 - s 8


 
871 . Coloring, {3, 6, 7, 10, 11}

R: [7, 7, 8, 6, A, 3, A, C, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y7, y5, y6, y4, y3, 0, y10, y8, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y7 - y6 - y4 - y3 + y2 + y1, 0, y5, 0, 0, y7, y6, y5, y4, y3, y2, y1]

  p' = s 7 - s 8   p = s 7 - s 9


 
872 . Coloring, {3, 6, 7, 10, 12}

R: [7, 7, 8, 6, A, 3, A, C, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y10, y9, y8, 0, 0, y6, y7, y5, y3, y4, y2, y1]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y7, y8, y7, 0, y5, y6, y4]

  p = - s 6 + s 9


 
873 . Coloring, {3, 6, 7, 11, 12}

R: [7, 7, 8, 6, A, 3, A, C, B, C, 4, 9]
B: [6, 8, 7, 7, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y7, y8, y9, y5, y6]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 + y6 - y4 - y5 - y7 + y8 + y9, y1, y2, 0, y3, y6, y4, y5, 0, y7, y8, y9]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
874 . Coloring, {3, 6, 8, 9, 10}

R: [7, 7, 8, 6, A, 3, B, B, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y7 - y4 + y5, 0, 2 y7 - y4 + y5, y7, y3, y4, 0, y5, y6, y7]

  p' = - s 5 + s 8   p' = - s 6 + s 9   p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
875 . Coloring, {3, 6, 8, 9, 11}

R: [7, 7, 8, 6, A, 3, B, B, C, C, 4, 5]
B: [6, 8, 7, 7, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 3 y5, 3 y6, 3 y7, 3 y8, 3 y4, 3 y2, 0, 3 y3, -3 y5 - 3 y6 + 5 y7 - 3 y8 - 3 y4 - 3 y2 + 5 y3 + 5 y1, 3 y1]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y5, y6, y7, 0, 0, y8, y1, y2, y3, y4, y9, y10]


 
876 . Coloring, {3, 6, 8, 9, 12}

R: [7, 7, 8, 6, A, 3, B, B, C, C, 1, 9]
B: [6, 8, 7, 7, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, 0, -3 y1 - 3 y3 - 3 y2 - 3 y4 - 3 y6 + 10 y5, 0, 0, 3 y4, 3 y3, 3 y2, -3 y4 + 3 y5, 3 y4, 3 y6, 3 y5]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y3, y4, y2, y1, y6, y5, y9, 0, y8, y6, y7]

  p = s 3 - s 10


 
877 . Coloring, {3, 6, 8, 10, 11}

R: [7, 7, 8, 6, A, 3, B, B, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 5 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y5, 0, y5, 0, 0, -3 y5 + 2 y1, y1, y5, y2, y4, 0, y3]

  p = s 4 - s 8   p' = s 5 - s 7   p' = - s 4 + s 6


 
878 . Coloring, {3, 6, 8, 10, 12}

R: [7, 7, 8, 6, A, 3, B, B, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y2 - y5, 0, 0, y5, y3, y4, 2 y5, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, 2 y5, y3, y5, y1, y5, 0, y6, 0, y4]

  p = - s 2 + s 7   p' = - s 2 + s 7


 
879 . Coloring, {3, 6, 8, 11, 12}

R: [7, 7, 8, 6, A, 3, B, B, B, C, 4, 9]
B: [6, 8, 7, 7, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y6, y5, 0, y3, 2 y1, y4, y2, y1, y7, y8]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y3, y2, y1, 0, y8, y7, y6, y5, 0, y4, 0, y9]


 
880 . Coloring, {3, 6, 9, 10, 11}

R: [7, 7, 8, 6, A, 3, B, C, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, y3, y2, y1, -y3 + y2 + y1 - y9 + y7 - y8 + y6 - y5 + y4, y9, y7, y8, 0, y6, y5, y4]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
881 . Coloring, {3, 6, 9, 10, 12}

R: [7, 7, 8, 6, A, 3, B, C, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y2, 7 y2 + 7 y5 + 7 y1 - 9 y7 - 9 y6 + 7 y4 - 9 y3, 7 y2 + 5 y5 + 7 y1 - 9 y7 - 9 y6 + 7 y4 - 9 y3, 0, 0, 2 y5, 2 y1, 2 y7, 2 y6, 2 y5, 2 y4, 2 y3]

  p = - s 4 + s 10   p = - s 4 + s 6 + s 7 - s 9   p = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y8, y7, y5, y6, y4, y6, 0, y3, y2, y1]

  p = s 4 - s 9


 
882 . Coloring, {3, 6, 9, 11, 12}

R: [7, 7, 8, 6, A, 3, B, C, C, C, 4, 9]
B: [6, 8, 7, 7, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y2, y1, 0, y8, 2 y6, y7, y5, y6, y4, y3]

  p = s 7 - s 9

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, 0]


 
883 . Coloring, {3, 6, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, B, C, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y9, y10, y8, 0, y7, y6, y4, y5, y3, y1, y2]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y3, 0, y4, 0, y1, y2, y8, y9, 0, y5, y6, y7]


 
884 . Coloring, {3, 7, 8, 9, 10}

R: [7, 7, 8, 6, A, A, A, B, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y6 + y5, y6, y3, y6, 0, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -y2 + y6, y3, 0, y2, y1, y2, y4, 0, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
885 . Coloring, {3, 7, 8, 9, 11}

R: [7, 7, 8, 6, A, A, A, B, C, C, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, y3, y4, 2 y1, y1, 0, y2, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y9, y5, y6, 0, y7, y8]


 
886 . Coloring, {3, 7, 8, 9, 12}

R: [7, 7, 8, 6, A, A, A, B, C, C, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, y6, y7, y5, y4, y3, y2]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y7, y8, y6, y5, y7, y4, y3, 0, 0, y1, y2]

  p = s 6 - s 9


 
887 . Coloring, {3, 7, 8, 10, 11}

R: [7, 7, 8, 6, A, A, A, B, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, y3, y7, y4, 0, y5, y6, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-7 y1 - 7 y2 - 7 y3 + 9 y4 + 9 y5 - 7 y6 + 9 y7, 0, 7 y1, 0, 0, 7 y2, 7 y3, 7 y4, 7 y5, 0, 7 y6, 7 y7]

  p = - s 2 - s 3 + s 7 + s 8


 
888 . Coloring, {3, 7, 8, 10, 12}

R: [7, 7, 8, 6, A, A, A, B, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y5, y6, 0, 0, 0, y4, y3, y4, 2 y4, y2, y1, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y5, y4, y5, 0, 0, y6, y7]

  p = s 5 - s 8


 
889 . Coloring, {3, 7, 8, 11, 12}

R: [7, 7, 8, 6, A, A, A, B, B, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, 0, y8, y9]


 
890 . Coloring, {3, 7, 9, 10, 11}

R: [7, 7, 8, 6, A, A, A, C, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, 0, 2 y5, y1, -3 y5 + 2 y3, y6, y5, 0, y4, 0, y3]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, 0, y7, 2 y5]

  p = - s 3 + s 8


 
891 . Coloring, {3, 7, 9, 10, 12}

R: [7, 7, 8, 6, A, A, A, C, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-10 y1 - 10 y2 + 22 y3 - 10 y4 + 22 y5, 4 y1, 0, 0, 0, -5 y1 - 5 y2 + 11 y3 - 5 y4 + 11 y5, 4 y2, -5 y1 - 5 y2 + 11 y3 - 5 y4 + 11 y5, 4 y3, 4 y4, 0, 4 y5]

  p' = s 3 + s 4 - s 6 - s 7   p = s 2 - s 8   p' = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y6, y5, y4, y3, y2, y3, 0, 0, y1, 2 y3]

  p' = s 4 - s 7   p = s 4 - s 7


 
892 . Coloring, {3, 7, 9, 11, 12}

R: [7, 7, 8, 6, A, A, A, C, C, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y3, 0, y5, 2 y3, y3, y4, y1, 0, y2]

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y5, y4, y3, 0, y4, y2, y1, y7, 0, 0, y6, 0]

  p = s 3 - s 8


 
893 . Coloring, {3, 7, 10, 11, 12}

R: [7, 7, 8, 6, A, A, A, C, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y4, 0, y3, 0, y2, y1, y9, y8, y7, y6, y5]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y2, 0, 0, y1, 3 y2]

  p = - s 3 + s 8


 
894 . Coloring, {3, 8, 9, 10, 11}

R: [7, 7, 8, 6, A, A, B, B, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y6, 0, y5, y4, y3, y2, y7, 0, y1, -y6 + y5 - y4 - y3 + y2 + 2 y7 + y1, y7]

  p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = s 3 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y4, 0, y3, 0, 0, y1, y2, y9, y7, y8, y6, y5]


 
895 . Coloring, {3, 8, 9, 10, 12}

R: [7, 7, 8, 6, A, A, B, B, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 6 y3 - 3 y2 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 3 y1, 0, 0, 0, 3 y3, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y7, y5, y4, y6, y3, y6, 0, y2, y6, y1]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
896 . Coloring, {3, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s + s 2 + 3s 4 + 4s 5 + 8s 7 + 16s 8

R: [7, 7, 8, 6, A, A, B, B, C, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, y6, 2 y5, y5, y4, y3, y2, y1]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y8, y9, y10, y4, 0, y5, y6, y7]


 
897 . Coloring, {3, 8, 10, 11, 12}

R: [7, 7, 8, 6, A, A, B, B, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y4, 0, y3, 0, y2, y1, y7, 2 y7, y6, y5, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y5, 0, y2, 0, y1, y6, y7, y5, 0, y4, 0, y3]

  p = - s 3 + s 8


 
898 . Coloring, {3, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, A, B, C, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y1, 0, 8 y1 + 8 y2 - 3 y3 - 3 y6 + 8 y5 - 11 y7, 0, 3 y2, 3 y3, 5 y1 + 5 y2 - 3 y4 + 5 y5 - 8 y7, 3 y4, 3 y6, 3 y5, 3 y7]

  p = - s 2 + s 8   p' = s 2 - s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y5, 0, y4, y2, y3, y8, 0, y9, y7, y6]


 
899 . Coloring, {4, 5, 6, 7, 8}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7   p = 3s 2 + 14s 3 - 24s 5 + 32s 6 - 32s 7 - 128s 8

R: [7, 7, 7, 7, 3, 3, A, B, B, C, 1, 5]
B: [6, 8, 8, 6, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y7 - y6 + y5 + y4 + y3 - y2 - y1, 0, y7, 0, y6, 0, y5, y4, y3, y2, y1]

  p = s 7 - s 8


 
900 . Coloring, {4, 5, 6, 7, 9}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, A, C, C, C, 1, 5]
B: [6, 8, 8, 6, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y5, 0, 0, y6, 0, y4]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y2 + y3 + y4 + y5 - y6, 0, y1, 0, y2, 0, y3, y4, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
901 . Coloring, {4, 5, 6, 7, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, 2, 1, 5]
B: [6, 8, 8, 6, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, 0, y4, 0, 0, y5, y6, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y2 + y1 + y4 + y5 - y6 - y3, 0, y1, y4, y5, y6, y3]

  p = s 6 - s 7


 
902 . Coloring, {4, 5, 6, 7, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, C, 4, 5]
B: [6, 8, 8, 6, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, 0, 0, y6, y7, y5]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y1 + 5 y2 - 3 y3 + 8 y5, 3 y4, 0, 0, 0, -3 y4 + 8 y2 - 3 y6 + 5 y5, 0, 3 y1, 3 y2, 3 y3, 3 y6, 3 y5 ]

  p = - s + s 7   p' = - s + s 7


 
903 . Coloring, {4, 5, 6, 7, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, C, 1, 9]
B: [6, 8, 8, 6, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y2 - y3 - y4 + y5 + y6, 0, y1, 0, 0, 0, y2, 0, y3, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y4, 0, y5, 0, y6, y7, y8]


 
904 . Coloring, {4, 5, 6, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, C, 1, 5]
B: [6, 8, 8, 6, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, 0, 0, 0, y5, y6]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y2, 0, y3, 0, y4, y5, y6, y7, y8]


 
905 . Coloring, {4, 5, 6, 8, 10}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, B, B, B, 2, 1, 5]
B: [6, 8, 8, 6, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y3, y2, 0, y3, 0, y4, 0, 0, 0, y5, 0]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, y4, 0, y5, 0, y4, y3, y2, 0, y1]

  p = - s 4 + s 6


 
906 . Coloring, {4, 5, 6, 8, 11}

Ωp(Δ)=0:     p' = s 2 - 8s 5 - 16s 6   p' = s 3 - 8s 6 - 16s 7   p = s 2 - 8s 5 - 16s 6

R: [7, 7, 7, 7, 3, 3, B, B, B, C, 4, 5]
B: [6, 8, 8, 6, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 6 vs 6 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y3, y2, 0, y4, 0, 0, 0, y6, y5]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, 0, y4, y5, y6, 0, y7]


 
907 . Coloring, {4, 5, 6, 8, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, B, B, B, C, 1, 9]
B: [6, 8, 8, 6, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y4, 0, y5, 0, 0, 0, y1, 0, y2, 0, y3, y5]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, y4, 0, y5, 0, y6, 0, y7]


 
908 . Coloring, {4, 5, 6, 9, 10}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, B, C, C, 2, 1, 5]
B: [6, 8, 8, 6, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y6, y1, 0, y4, 0, y3, 0, 0, 0, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, y1, 0, y2, y4, y5, y6, y7]


 
909 . Coloring, {4, 5, 6, 9, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, 3, B, C, C, C, 4, 5]
B: [6, 8, 8, 6, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y4, 0, y5, 0, 0, 0, y3, y6]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, 0, 0, 0, y5, 0, y4, y3, y2, y1, 0]


 
910 . Coloring, {4, 5, 6, 9, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, B, C, C, C, 1, 9]
B: [6, 8, 8, 6, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, 0, -3 y1 - 3 y2 + 5 y5 - 3 y4 + 5 y3, 0, 0, 0, 3 y2, 0, 3 y5, 0, 3 y4, 3 y3]

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y2 - y3 + y4 + y5 - y6, 0, y1, y2, y3, 0, y4, 0, y5, y6, 0]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
911 . Coloring, {4, 5, 6, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, 2, 4, 5]
B: [6, 8, 8, 6, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y6, y4, y1, y2, 0, y3, 0, 0, 0, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, 0, y6, y4, y5, y3, y2]


 
912 . Coloring, {4, 5, 6, 10, 12}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, 2, 1, 9]
B: [6, 8, 8, 6, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 2 y5, 2 y5, 0, 0, 0, y2, 0, y4, 0, y3, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y6, y5, 0, y4, 0, y3, y2, y1]


 
913 . Coloring, {4, 5, 6, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, C, 4, 9]
B: [6, 8, 8, 6, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 2 y1, 2 y2, 0, 0, 2 y5, 0, 2 y4, 0, 2 y3, 3 y1]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y8, y6, 0, y7, 0, y3, y4, y5]


 
914 . Coloring, {4, 5, 7, 8, 9}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, A, B, C, C, 1, 5]
B: [6, 8, 8, 6, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, 0, y4, 0, y5, y6, y7, y8, y9]


 
915 . Coloring, {4, 5, 7, 8, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y1, y2, y3, 0, 2 y1 - 2 y3, 0, y4, 0, 0, y5, 2 y1 - 2 y3, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y3, y2, 0, y1, 0, y6, y5, y4, 2 y4, y7]

  p = s 6 - s 8


 
916 . Coloring, {4, 5, 7, 8, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, C, 4, 5]
B: [6, 8, 8, 6, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, y3, y2, 0, y1, 0, 0, y7, y5, y6]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2 - 2 y6, y2, y1, 0, 0, y4, 0, y3, y7, y6, 2 y6, y5]

  p' = - s 6 + s 8   p = - s 6 + s 8


 
917 . Coloring, {4, 5, 7, 8, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, C, 1, 9]
B: [6, 8, 8, 6, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y2 - y3 - y4 + y5 + y6, 0, y1, 0, 0, 0, y2, 0, y3, y4, y5, y6]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y2, y1, y7, y8, y6, 0, y5, 0, y3, y4, y9]


 
918 . Coloring, {4, 5, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, y4, 0, y3, 0, y2, 0, 0, y1, 0, y6]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y3, y2, 0, y1, 0, y7, y8, y6, y4, y5]


 
919 . Coloring, {4, 5, 7, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, C, 4, 5]
B: [6, 8, 8, 6, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, 0, 0, y5, 0, y6]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y3, y4, y5, 0, 0, y6, 0, y7, 2 y1, y1, y2, 0]

  p = - s 3 + s 8


 
920 . Coloring, {4, 5, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, C, 1, 9]
B: [6, 8, 8, 6, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 6 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, y4, 0, 0, 0, y3, 0, y1, y2, 0, 3 y4 - y3 + y1 + y2]

  p' = - s 4 + s 5   p = s 4 - s 5

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y5, y1, y2, y3, y4, 0, y6, 0, y7, y8, 0]


 
921 . Coloring, {4, 5, 7, 10, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, A, C, B, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, y4, y2, y2, 0, y1, 0, 0, y5, y6, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[-3 y1 - 3 y3 - 3 y2 - 3 y5 + 10 y6, 0, 3 y1, 0, 0, 3 y3, 0, 3 y2, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p' = - s 2 + s 7   p = - s 2 + s 7


 
922 . Coloring, {4, 5, 7, 10, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, A, C, B, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, 0, 0, y7, 0, y5, y6, y4, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 3 y2, 3 y3, 3 y1, -3 y2 - 3 y3 + 5 y1 - 3 y4 + 5 y5 - 3 y6 + 5 y7, 0, 3 y4, 0, 3 y5, 3 y6, 3 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
923 . Coloring, {4, 5, 7, 11, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, A, C, B, C, 4, 9]
B: [6, 8, 8, 6, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 - y6 - y3 + y4 + y5, y1, 0, 0, y2, 0, y6, y3, y4, y5]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y8, y7, 0, y6, y5, 0, y4, 0, y3, y2, y9]


 
924 . Coloring, {4, 5, 8, 9, 10}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, B, C, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y3, y2, 0, y3, 0, y4, 0, 0, y6, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y6, y7, 0, y5, 0, y3, y4, y2, y1, y8]


 
925 . Coloring, {4, 5, 8, 9, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, A, B, B, C, C, 4, 5]
B: [6, 8, 8, 6, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y2, y1, 0, y7, 0, 0, y6, y5, y4]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y3, y4, y2, 0, 0, y1, 0, y6, y5, y8, y9, y7]


 
926 . Coloring, {4, 5, 8, 9, 12}

Ωp(Δ)=0:     p' = - s 2 - 2s 3 + 8s 4 + 8s 5 - 32s 7   p = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, B, C, C, 1, 9]
B: [6, 8, 8, 6, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y4, 0, 3 y3, 0, 0, 0, 3 y2, 0, 3 y1, 3 y3, -3 y4 + 7 y3 - 3 y2 + 10 y1, 3 y3 + 3 y1]

  p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y2, y1, y9, y8, y7, 0, y6, 0, y5, y4, y3]


 
927 . Coloring, {4, 5, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, B, B, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y4, 2 y4 - 3 y1, y3, 2 y1, 0, y2, 0, 0, y1, y5, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y1, 0, 2 y2, 0, 0, 2 y3, 0, 2 y4, 2 y5, 3 y1, 0, 2 y6]

  p = - s 5 + s 7


 
928 . Coloring, {4, 5, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, B, B, B, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y2, y1, y4, 0, 0, 0, y5, 0, 2 y4, y4, y3, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, 0, y5, 0, y6, 0, y7]


 
929 . Coloring, {4, 5, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, B, B, C, 4, 9]
B: [6, 8, 8, 6, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, 0, y2, 0, y6, y4, y5, y3]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, 0, y6, 0, y7, 0, y8]


 
930 . Coloring, {4, 5, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 2 y1 - 2 y6, 0, y5, 0, 0, y6, y4, 2 y6]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y8, 0, y7, 0, 0, y6, 0, y5, y4, y3, y2, y1]


 
931 . Coloring, {4, 5, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, 0, -y1 - y2 - 2 y3 + 6 y5 - y4, 0, y5, y3, y4, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 7 y1, -7 y1 + 9 y3 - 7 y4 - 7 y2 + 9 y5 - 7 y6 + 9 y7, 7 y3, 7 y4, 0, 7 y2, 0, 7 y5, 7 y6, 7 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
932 . Coloring, {4, 5, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, C, 4, 9]
B: [6, 8, 8, 6, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 7 y1 + 7 y2 - 9 y5 + 7 y3 - 9 y4, 2 y1, 0, 0, 2 y2, 0, 2 y5, 7 y1 + 7 y2 - 9 y5 + 7 y3 - 9 y4, 2 y3, 2 y4]

  p = s 2 - s 4 - s 5 + s 7   p' = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y3, y1, y2, 0, y5, y4, 0, y8, 0, y7, y6, 0]


 
933 . Coloring, {4, 5, 10, 11, 12}

Ωp(Δ)=0:     p' = - 3s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7   p = 3s 2 + 10s 3 + 40s 5 + 32s 6 + 32s 7 + 128s 8

R: [7, 7, 7, 7, 3, A, B, C, B, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y5, y4, y3, 0, 0, y2, 0, y5, y4, y1, y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[y4, 0, y3, 0, y2, y1, 0, -y4 - y3 + y2 - y1 + y7 - y6 + y5, 0, y7, y6, y5]

  p = s + s 2 + s 3 - s 6 - s 7 - s 8


 
934 . Coloring, {4, 6, 7, 8, 9}

Ωp(Δ)=0:     p = 9s 2 - 4s 4 + 24s 5 - 16s 6 + 96s 7 - 64s 8   p' = 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, A, B, C, C, 1, 5]
B: [6, 8, 8, 6, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y2, 0, y3, 0, 0, y4, y5, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y7, y8, y9, 0, y1, 0, y2, y3, y4, y5, y6]


 
935 . Coloring, {4, 6, 7, 8, 10}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y3, y2, y1, 0, 2 y1, 0, y5, 0, 0, y4, 2 y1, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, -2 y1 + 2 y3, 0, y2, 0, y3, y6, y4, 2 y1, y5]

  p = - s 5 + s 7   p' = - s 5 + s 7


 
936 . Coloring, {4, 6, 7, 8, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, C, 4, 5]
B: [6, 8, 8, 6, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y4, 0, y3, 0, 0, y5, 2 y1, y6]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y2, y6, y1, 0, 0, y5, 0, y4, y3, y7, 2 y1, y8]

  p = s 7 - s 9


 
937 . Coloring, {4, 6, 7, 8, 12}

Ωp(Δ)=0:     p = - s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, C, 1, 9]
B: [6, 8, 8, 6, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y2 + y1 - y5 - y6 + y4 + y3, 0, y2, 0, 0, 0, y1, 0, y5, y6, y4, y3]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y9, y7, y8, y6, y5, 0, y4, 0, y3, y2, y1]


 
938 . Coloring, {4, 6, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[2 y5, y4, y5, 0, y1, 0, y2, 0, 0, y3, 0, 2 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y2, 0, y8, y6, y7, y4, y5]


 
939 . Coloring, {4, 6, 7, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, C, 4, 5]
B: [6, 8, 8, 6, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y1, 2 y1, y2, 0, y4, 0, 0, y3, 0, y5]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, 0, y4, 0, y3, 2 y5, y2, y1, 0]

  p = - s 2 + s 8


 
940 . Coloring, {4, 6, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, C, 1, 9]
B: [6, 8, 8, 6, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 6 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, y1, 0, 0, 0, 3 y1 + y2 + y3 - y4, 0, y2, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y4, y5, y3, y1, y2, 0, y8, 0, y7, y6, 0]


 
941 . Coloring, {4, 6, 7, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, A, C, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y5, y3, y3, 0, y2, 0, 0, y4, y5, y5]

  p' = s 4 - s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y8, 0, y7, 0, 0, y6, 0, y4, y5, y3, y1, y2]


 
942 . Coloring, {4, 6, 7, 10, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, A, C, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y5, y1, y7, 0, 0, 0, y2, 0, y3, y4, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y5, 0, y4, 0, y8, y6, y7]


 
943 . Coloring, {4, 6, 7, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, A, 3, A, C, B, C, 4, 9]
B: [6, 8, 8, 6, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 - y3 - y4 + y6 + y5, y1, 0, 0, y2, 0, y3, y4, y6, y5]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y7, y6, 0, y5, y4, 0, y3, 0, y2, y9, y8]


 
944 . Coloring, {4, 6, 8, 9, 10}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, A, 3, B, B, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y6, 0, y4, 0, y3, 0, 0, y5, y7, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y4, 0, y2, 0, y3, y5, y6, y8, y7]


 
945 . Coloring, {4, 6, 8, 9, 11}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, A, 3, B, B, C, C, 4, 5]
B: [6, 8, 8, 6, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y2, y1, y4, 0, 4 y2 + 4 y1 + 5 y4 - 6 y3, 0, 0, -2 y2 - 2 y1 - 4 y4 + 5 y3, y3, 5 y2 + 5 y1 + 6 y4 - 8 y3]

  p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y5, y4, 0, 0, y3, 0, y2, y8, y9, y7, y6]


 
946 . Coloring, {4, 6, 8, 9, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, B, B, C, C, 1, 9]
B: [6, 8, 8, 6, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-3 y2 - 3 y1 + 10 y4 - 3 y3, 0, 3 y2, 0, 0, 0, 3 y1, 0, -3 y2 + 3 y4, 3 y2, 3 y3, 3 y4]

  p = s 2 - s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y5, y2, y3, y4, 0, y6, 0, y7, y8, y9]


 
947 . Coloring, {4, 6, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, B, B, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, y3, y2, 2 y3, 0, y1, 0, 0, y6, y5, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y5, 0, y5, 0, 0, -2 y5 + 2 y3, 0, y3, y4, y2, 0, y1]

  p = - s 4 + s 6   p' = - s 4 + s 6


 
948 . Coloring, {4, 6, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, B, B, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y2, y3, y5, 0, 0, 0, y4, 0, 2 y5, y5, y1, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y3, y1, y2, y7, 0, y6, 0, y5, 0, y4]


 
949 . Coloring, {4, 6, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 24s 6 - 16s 7 - 64s 8   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 7, 7, 7, A, 3, B, B, B, C, 4, 9]
B: [6, 8, 8, 6, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y6, 0, 0, y4, 0, y3, y5, y2, y1]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y4, y3, y2, 0, y1, y6, 0, y5, 0, y8, 0, y7]


 
950 . Coloring, {4, 6, 9, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y1, y5, y4, 0, y3, 0, 0, y7, y6, 2 y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y6, 0, y4, y5, y3, y8, y7]


 
951 . Coloring, {4, 6, 9, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y1, -y1 - 2 y4 - y3 + 6 y5 - y2, y4, 0, 0, 0, y3, 0, y5, y4, y2, y5]

  p' = s 3 - s 6   p' = s 4 - s 7   p = s 3 - s 6

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4, y3, y2, y1, 0, y7, 0, y6, y5, y8]


 
952 . Coloring, {4, 6, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, C, 4, 9]
B: [6, 8, 8, 6, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 7 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y5, 0, 0, 2 y4, 0, 2 y3, 7 y5 + 7 y4 - 9 y3 + 7 y2 - 9 y1, 2 y2, 2 y1]

  p' = s 2 + s 3 - s 5 - s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y8, y7, y6, 0, y5, y4, 0, y3, 0, y2, y1, 0]


 
953 . Coloring, {4, 6, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 - 2s 3 - 8s 5 - 32s 7   p' = - s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, B, C, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y5, y4, y1, 0, 0, y2, 0, y5, y4, y3, y4]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y8, 0, y7, y6, 0, y5, 0, y4, y3, y2]


 
954 . Coloring, {4, 7, 8, 9, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2, y1, 0, 0, y2, 0, y3, 0, 0, y4, y5, y5]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y7, 0, y6, 0, y5, y4, 0, y3, y2]


 
955 . Coloring, {4, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, C, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 8 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, 0, y6, 0, 0, y3, y4, y5]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y2, y1, y8, 0, 0, y7, 0, y6, y5, 0, y4, y3]


 
956 . Coloring, {4, 7, 8, 9, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, C, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y3 - y4 - y5 + y1 + y2, 0, 0, 0, 0, 0, y3, 0, y4, y5, y1, y2]

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 2 y1, 2 y2, 2 y3, 2 y4, 2 y5, 0, 2 y7, 0, 0, 3 y1, 2 y6]

  p = - s 4 + s 8


 
957 . Coloring, {4, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, A, A, B, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, y2, y5, 0, y3, 0, 0, y4, y5, 0]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, y2 + y1 - y4 - y3 - y6 + y5, 0, y4, y3, 0, y6, y5]

  p = s 6 - s 7


 
958 . Coloring, {4, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, B, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, 0, y4, y5, y6, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y4, y3, y2, y1, 0, y4 + y3 - y2 - y1 - y6 + y5, 0, 0, y6, y5]

  p = s 4 - s 5 + s 6 - s 7


 
959 . Coloring, {4, 7, 8, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7   p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, A, A, A, B, B, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 4 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5 - y3 - y4 + y2 + y1, 0, 0, y5, 0, y3, y4, y2, y1]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y3 + y4 - y2, y3, y4, 0, y1, y3 + y4 - y1, 0, y4, 0, 0, y3, y2]

  p' = s 4 - s 5   p' = - s 5 + s 7   p = s 4 - s 6   p' = - s 5 + s 6


 
960 . Coloring, {4, 7, 9, 10, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, A, A, A, C, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, y4, y5, 0, y3, 0, 0, y2, 0, y4]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y4, y5, 0, y6, y7]


 
961 . Coloring, {4, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, C, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 2 vs 6 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-y1 + 2 y2, y1, 0, 0, 0, 0, 2 y2, 0, y2, 2 y2, 0, y2]

  p = - s 2 + s 3   p = - s 2 + s 4   p = - s 2 + s 5   p = - s 2 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y5, y4, y2, y3, 0, y1, 0, 0, y7, y6]


 
962 . Coloring, {4, 7, 9, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, A, A, A, C, C, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 5 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, y4, 0, 0, y3, 0, y1, y2, 0, y4 - y3 + y1 + y2]

  p = - s 4 + s 5

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y3, y2, 0, y3, y4, 0, y5, 0, 0, y6, 0]

  p = - s 2 + s 7


 
963 . Coloring, {4, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, C, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, 0, y4, y6, y5, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y3, y4, 0, y5, 0, 0, y6, y7]


 
964 . Coloring, {4, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, y7, 0, y6, 0, 0, y5, y4, y3]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y4, y5, y6, y7, y8]


 
965 . Coloring, {4, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y6, 3 y5, 0, 0, 0, 0, 3 y4, 0, 3 y3, 3 y2, -3 y6 - 3 y5 - 3 y4 + 13 y3 - 3 y2 + 13 y1, 3 y1]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y1, y3, y4, y2, 0, y6, 0, 2 y5, y5, y7]

  p = s 4 - s 8


 
966 . Coloring, {4, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, -7 y5 + 9 y4 + 9 y3 - 7 y2 + 9 y1, 0, 0, 7 y5, 0, 7 y4, 7 y3, 7 y2, 7 y1]

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y1, 2 y1 - 2 y6, y2, 0, y4, y3, 0, y5, 0, 2 y6, y6, y7]

  p' = - s 4 + s 8   p = - s 4 + s 8


 
967 . Coloring, {4, 8, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 7, 7, 7, A, A, B, B, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, 0, y4, y4, y5, 0]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y4, 0, y1 + y2 + y3 - y5, 0, y1, y2, 0, y3, 0, y4, 0, y5]

  p = - s 3 + s 7   p = - s 3 + s 4 - s 5 + s 6


 
968 . Coloring, {4, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 - 8s 5 + 32s 7   p = - 9s 2 + 4s 4 - 24s 5 - 16s 6 + 96s 7 + 64s 8

R: [7, 7, 7, 7, A, A, B, C, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 3 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -y1 + 2 y3, 0, y1, 0, 0, 2 y3, 0, y3, -y2 + 2 y3, y2, y3]

  p = s 3 - s 7   p' = s 4 - s 6   p' = s 5 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, 0, y3, 0, y1, y8, 0, y7, 0, y6, y5, y4]


 
969 . Coloring, {5, 6, 7, 8, 9}

R: [7, 7, 7, 6, 3, 3, A, B, C, C, 1, 5]
B: [6, 8, 8, 7, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y3, 0, y2, y6, y4, 0, 0, y5, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, 0, y2, 0, y5, y6, y7, y4, y3, y8, y9]


 
970 . Coloring, {5, 6, 7, 8, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, 2, 1, 5]
B: [6, 8, 8, 7, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[2 y1, 2 y2, 3 y1 - 2 y3, 0, 4 y3, 2 y3, 2 y4, 0, 0, 2 y5, 4 y3, 0]

  p' = s 3 - s 6   p' = s 4 - s 7   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, 5 y3, 0, 5 y1, 5 y2, 10 y1, 11 y3 - 15 y1 + 11 y2 - 5 y4 + 11 y5 - 5 y6, 5 y4, 5 y5, 5 y6]

  p = - s 3 + s 5 + s 6 - s 8   p = s 3 + s 4 - s 6 - s 7


 
971 . Coloring, {5, 6, 7, 8, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, C, 4, 5]
B: [6, 8, 8, 7, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, y1, y2, y5, y3, 0, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y2, y1, 0, 0, 0, y3, y4, y6, y5, y9, y8, y7]


 
972 . Coloring, {5, 6, 7, 8, 12}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, C, 1, 9]
B: [6, 8, 8, 7, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y3 + y2 - y5 - y4 + y6 + y7, 0, y1, 0, 0, y3, y2, 0, y5, y4, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, 11 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y7 - 5 y8, 0, 5 y1, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
973 . Coloring, {5, 6, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, 2, 1, 5]
B: [6, 8, 8, 7, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y1, y4, y3, 0, y2, y1, y6, 0, 0, y5, 0, 2 y1]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, y1, y2, 2 y1, y4, y5, y6, y7]

  p = s 5 - s 8

 ยป SYNC'D !RANK'D


 
974 . Coloring, {5, 6, 7, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, C, 4, 5]
B: [6, 8, 8, 7, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 0, y7, y6, y5, 2 y6, y4, y3, 0]

  p = s 2 - s 8


 
975 . Coloring, {5, 6, 7, 9, 12}

Ωp(Δ)=0:     p = - 2s 2 + 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, C, 1, 9]
B: [6, 8, 8, 7, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y4, 0, y5, 0, 0, y4, y3, 0, y2, y1, 0, y5 + y4 - y3 + y2 + y1]

  p = s 5 - s 7   p' = s 5 - s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y3, 2 y2, y2, y5, y6, 0, y7, y4, 0]

  p = - s 5 + s 8


 
976 . Coloring, {5, 6, 7, 10, 11}

R: [7, 7, 7, 6, 3, 3, A, C, B, 2, 4, 5]
B: [6, 8, 8, 7, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y7, y6, y5, y5, y4, y3, 0, 0, y2, y1, y1]

  p' = s 5 - s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, 2 y3, y4, y7, y5, y6]

  p = s 6 - s 8


 
977 . Coloring, {5, 6, 7, 10, 12}

R: [7, 7, 7, 6, 3, 3, A, C, B, 2, 1, 9]
B: [6, 8, 8, 7, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y7, y6, y5, 0, 0, y4, y3, 0, y5, y2, y1, y4]

  p' = s 5 - s 8   p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y1, y2, -y2 + y4, 2 y2 - 2 y4 + y3, -2 y2 + 2 y4, 0, y1, y4, y3]

  p = s 2 - s 5   p' = s 2 - s 5   p' = - s 4 + s 7   p' = - s 3 + s 6


 
978 . Coloring, {5, 6, 7, 11, 12}

R: [7, 7, 7, 6, 3, 3, A, C, B, C, 4, 9]
B: [6, 8, 8, 7, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1 - y5 + y4, y1 + y4 - y3, 0, y1 + y4 - y2, y1, 0, y5, y3, y4, y2]

  p = s - s 2 + s 5 - s 6   p' = - s + s 2 - s 5 + s 6   p' = - s + s 3 - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y6 + y5 - y4 - y3 + y2 - 2 y7, y1, 0, 0, y6, y5, y7, y4, 0, y3, y2, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


 
979 . Coloring, {5, 6, 8, 9, 10}

R: [7, 7, 7, 6, 3, 3, B, B, C, 2, 1, 5]
B: [6, 8, 8, 7, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y6, y2, 0, y3, y6, y4, 0, 0, 0, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y7, 2 y2, y6, y4, y5, y3]

  p = - s 2 + s 8


 
980 . Coloring, {5, 6, 8, 9, 11}

R: [7, 7, 7, 6, 3, 3, B, B, C, C, 4, 5]
B: [6, 8, 8, 7, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y2, y1, y7, y6, y5, 0, 0, 0, y4, y3]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y6, y7, 0, 0, 0, y3, y4, y5, y2, y1, y8, y9]


 
981 . Coloring, {5, 6, 8, 9, 12}

R: [7, 7, 7, 6, 3, 3, B, B, C, C, 1, 9]
B: [6, 8, 8, 7, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-5 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 + 11 y6, 0, 5 y1, 0, 0, 5 y2, 5 y3, 0, 5 y4, 0, 5 y5, 5 y6]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y8, y7, y6, y5, y4, 0, y3, y6, y2]

  p = s 4 - s 9


 
982 . Coloring, {5, 6, 8, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, 2, 4, 5]
B: [6, 8, 8, 7, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y3, y1, y2, y3, y4, y5, 0, 0, 0, y6, 0]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y3, 0, 0, 0, 0, y4, y3, 2 y3, y2, y1, 0, y5]

  p' = s 4 - s 6   p = s 4 - s 6

 ยป SYNC'D !RANK'D


 
983 . Coloring, {5, 6, 8, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, 2, 1, 9]
B: [6, 8, 8, 7, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y3, 2 y5, y4, 0, 0, y5, y2, 0, 2 y5, 0, y1, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y4, y5, y4, y3, 2 y4, 0, y2, 0, y1]

  p' = s 3 - s 6   p = s 3 - s 6

 ยป SYNC'D !RANK'D


 
984 . Coloring, {5, 6, 8, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, C, 4, 9]
B: [6, 8, 8, 7, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y4, 0, y3, y2, 0, y5, 0, y6, y7]

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y4, y1, 0, 0, y2, y3, y4, y5, 0, y7, 0, y6]

  p = s 3 - s 8


 
985 . Coloring, {5, 6, 9, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, 2, 4, 5]
B: [6, 8, 8, 7, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y7, y5, y4, y3, y2, y1, 0, 0, 0, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, y2, 2 y2, y3, y4, y5, y6]

  p = - s 2 + s 8


 
986 . Coloring, {5, 6, 9, 10, 12}

Ωp(Δ)=0:     p = - 2s 2 + 7s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, 2, 1, 9]
B: [6, 8, 8, 7, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-y1 - 3 y2 - y3 + 6 y5 - y4, 2 y2, y1, 0, 0, y2, y3, 0, y5, 0, y4, y5]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y5, y4, 2 y5, 0, y3, y7, y6]

  p = - s 5 + s 8


 
987 . Coloring, {5, 6, 9, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, C, 4, 9]
B: [6, 8, 8, 7, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 3 y2, 3 y1, 0, -3 y2 - 3 y1 - 3 y6 + 5 y5 - 3 y4 + 5 y3, 3 y6, 0, 3 y5, 0, 3 y4, 3 y3]

  p = - s - s 2 + s 6 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + 3 y3 - y4 - y5 + y6, y1, 0, 0, 2 y3, y2, y3, y4, 0, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8

 ยป SYNC'D !RANK'D


 
988 . Coloring, {5, 6, 10, 11, 12}

R: [7, 7, 7, 6, 3, 3, B, C, B, 2, 4, 9]
B: [6, 8, 8, 7, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y6, y1, y2, 0, y3, y4, 0, y7, 0, y5, y6]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, y1, y5, y4, 2 y4, 0, y3, y7, y6]

  p = s 5 - s 8


 
989 . Coloring, {5, 7, 8, 9, 10}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, 2, 1, 5]
B: [6, 8, 8, 7, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, y4, 0, y6, y2, y3, 0, 0, y1, y2, y2]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y8, 0, y7, y6, y5, y4, y7, y3, y2]

  p = s 6 - s 9

 ยป SYNC'D !RANK'D


 
990 . Coloring, {5, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, C, 4, 5]
B: [6, 8, 8, 7, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y3, y4, y5, y6, y7, 0, 0, y1, y2, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, 0, y4, y7, y5, y6, y7, y9, y8]

  p = - s 3 + s 10


 
991 . Coloring, {5, 7, 8, 9, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, C, 1, 9]
B: [6, 8, 8, 7, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y3, y1 - y3 + y5 + y4 - y2, 0, y5, y4, y3, y2]

  p' = s 6 - s 7   p' = s 5 - s 7   p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, -3 y1 + 5 y2 - 3 y3 - 3 y8 + 5 y4 - 3 y5 - 3 y6 + 5 y9 - 3 y7, 3 y1, 3 y2, 3 y3, 3 y8, 3 y4, 3 y5, 0, 3 y6, 3 y9, 3 y7]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10


 
992 . Coloring, {5, 7, 8, 10, 11}

R: [7, 7, 7, 6, 3, A, A, B, B, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, -2 y2 + 2 y3, y4, y5, 0, 0, y6, -2 y2 + 2 y3, 0]

  p' = s 4 - s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y5 + y3 + y6 - y7, 0, y1, 0, 0, y2, y4, y5, y3, y4, y6, y7]

  p = s 7 - s 9   p' = s 7 - s 8


 
993 . Coloring, {5, 7, 8, 10, 12}

R: [7, 7, 7, 6, 3, A, A, B, B, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y4, 0, 0, y4, y3, 0, 2 y4, y6, y5, 0]

  p = s 4 - s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2, 3 y1, 0, -7 y6 + 8 y5 - 7 y4 + 14 y2 - 3 y1, -5 y6 + 7 y5 - 5 y4 + 7 y2, -7 y6 + 14 y5 - 7 y4 - 3 y3 + 8 y2]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
994 . Coloring, {5, 7, 8, 11, 12}

R: [7, 7, 7, 6, 3, A, A, B, B, C, 4, 9]
B: [6, 8, 8, 7, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y2, 0, y1, y3 + y2 - y1 + y6 + y7 - y5 - y4, 0, y6, y7, y5, y4]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]


 
995 . Coloring, {5, 7, 9, 10, 11}

R: [7, 7, 7, 6, 3, A, A, C, C, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y3, 2 y4 - 2 y5, y4, y5, y6, 0, 0, y1, 0, 2 y4 - 2 y5]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y8, y4, y7, y8, y6, y5]

  p = s 4 - s 9


 
996 . Coloring, {5, 7, 9, 10, 12}

R: [7, 7, 7, 6, 3, A, A, C, C, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 6 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[2 y1, y4, y1, 0, 0, y1, -y4 - 4 y1 + 6 y3 - y2, 0, y3, y2, 0, y3]

  p = - s 2 + s 8   p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 3 y1, 3 y2, 3 y4, -7 y1 - 7 y2 + 14 y4 + 8 y5 - 3 y6, -7 y1 - 7 y2 + 8 y4 - 3 y3 + 14 y5, 3 y3, 0, 3 y5, 3 y6, -5 y1 - 5 y2 + 7 y4 + 7 y5]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
997 . Coloring, {5, 7, 9, 11, 12}

R: [7, 7, 7, 6, 3, A, A, C, C, C, 4, 9]
B: [6, 8, 8, 7, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, -5 y1 + 2 y2, y2, 0, -8 y1 + 3 y2 - y3 + y4, y3, 0, y4]

  p = s 4 - s 5   p' = - s 4 + s 5   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y6, y4, y5, 0, 2 y2, y3, y2, y1, 0, y8, y7, 0]

  p = - s 4 + s 9


 
998 . Coloring, {5, 7, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, C, B, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y4, y7, 0, y8, y5, 0, y6, y2, y3, y4]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[-3 y1 + 5 y2 - 3 y3 - 3 y4 - 3 y5 + 5 y6 - 3 y7 + 5 y8, 0, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 0, 3 y6, 3 y7, 3 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
999 . Coloring, {5, 8, 9, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y5 + y2 - y3 + y4 + y6 - y7 - y8, y1, y5, y2, y3, y4, 0, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y2, 0, y1, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1000 . Coloring, {5, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 6 y2 - 3 y7 + 13 y6 - 3 y5 - 3 y3 + 13 y4, 3 y1, 3 y2, 0, 0, 3 y2, 3 y7, 0, 3 y6, 3 y5, 3 y3, 3 y4]

  p' = s 4 + s 5 - s 7 - s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y4, y4 + y6, y3, y6, y2, -y4 + 2 y6 + y2, 0, y1, y6, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

 ยป SYNC'D !RANK'D


 
1001 . Coloring, {5, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, C, 4, 9]
B: [6, 8, 8, 7, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y3, y1, 0, y2, y3 - y1 + y2 - y5 - y6 + y7 + y4, 0, y5, y6, y7, y4]

  p = s 7 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y6, y4, y8, y5, 0, y7, y8, y9]

  p = - s 5 + s 10

 ยป SYNC'D !RANK'D


 
1002 . Coloring, {5, 8, 10, 11, 12}

R: [7, 7, 7, 6, 3, A, B, B, B, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y4 + y5 - y3 + y1 + y2 - y6, y4, y5, 0, y3, y1, 0, 2 y4, y2, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y3, 0, y4, 0, y1, y2, y3, y5, 0, y7, 0, y6]

  p = s 5 - s 8


 
1003 . Coloring, {5, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 + 3s 3 - 6s 4 + 8s 5 - 24s 6 + 32s 7 - 32s 8

R: [7, 7, 7, 6, 3, A, B, C, C, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -y1 - y2 + 3 y6 - y5, y1, -y3 + 3 y6 - y4, 0, y2, y3, 0, y6, y4, y5, y6]

  p' = - s 2 + s 8   p = - s 2 + s 8   p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[-y1 + y2 - y3 + y4 - y5 + y6 - y7 + y8, 0, y1, 0, y2, y3, y4, y5, 0, y6, y7, y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


M              \ ;       N

$ [ [0, 0, 0, 0, 168, 0, 78, 0, 0, 208, 0, 152] , [0, 0, 0, 152, 0, 0, 208, 0, 0, 81, 0, 165] , [0, 0, 0, 52, 94, 0, 170, 0, 0, 127, 0, 163] , [0, 152, 52, 0, 0, 78, 0, 0, 168, 0, 156, 0] , [168, 0, 94, 0, 0, 76, 0, 164, 0, 0, 104, 0] , [0, 0, 0, 78, 76, 0, 102, 0, 0, 162, 0, 188] , [78, 208, 170, 0, 0, 102, 0, 102, 240, 0, 312, 0] , [0, 0, 0, 0, 164, 0, 102, 0, 0, 236, 0, 104] , [0, 0, 0, 168, 0, 0, 240, 0, 0, 94, 0, 104] , [208, 81, 127, 0, 0, 162, 0, 236, 94, 0, 304, 0] , [0, 0, 0, 156, 104, 0, 312, 0, 0, 304, 0, 336] , [152, 165, 163, 0, 0, 188, 0, 104, 104, 0, 336, 0] ] $     $ [ [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0] , [1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0] , [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0] , [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0] , [0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0] ] $

τ= 74 , r'= 1/2

R: [7, 7, 7, 6, 3, A, B, C, C, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, A, B, B, C, 1, 5]

Ranges

Action of R on ranges, [[11], [25], [6], [24], [20], [25], [6], [24], [20], [11], [25], [6], [24], [21], [22], [14], [12], [13], [9], [30], [7], [28], [31], [31], [16], [8], [29], [8], [29], [5], [15]]
Action of B on ranges, [[21], [21], [22], [17], [23], [26], [27], [18], [23], [26], [26], [27], [18], [11], [25], [2], [12], [30], [3], [12], [13], [10], [30], [30], [3], [31], [19], [31], [19], [4], [1]]

Cycles:    R , {{2, 4, 6, 7, 10, 11}, {9, 12}},   B , {{1, 3, 6, 8, 11}, {5, 10, 12}}

β({1, 5}) = 7/202
β({1, 7}) = 13/808
β({1, 10}) = 13/303
β({1, 12}) = 19/606
β({2, 4}) = 19/606
β({2, 7}) = 13/303
β({2, 10}) = 27/1616
β({2, 12}) = 55/1616
β({3, 4}) = 13/1212
β({3, 5}) = 47/2424
β({3, 7}) = 85/2424
β({3, 10}) = 127/4848
β({3, 12}) = 163/4848
β({4, 6}) = 13/808
β({4, 9}) = 7/202
β({4, 11}) = 13/404
β({5, 6}) = 19/1212
β({5, 8}) = 41/1212
β({5, 11}) = 13/606
β({6, 7}) = 17/808
β({6, 10}) = 27/808
β({6, 12}) = 47/1212
β({7, 8}) = 17/808
β({7, 9}) = 5/101
β({7, 11}) = 13/202
β({8, 10}) = 59/1212
β({8, 12}) = 13/606
β({9, 10}) = 47/2424
β({9, 12}) = 13/606
β({10, 11}) = 19/303
β({11, 12}) = 7/101

Partitions
α([{1, 2, 3, 6, 8, 9, 11}, {4, 5, 7, 10, 12}]) = 1/1

b1 = {1, 2, 3, 6, 8, 9, 11} ` , ` b2 = {4, 5, 7, 10, 12}

Action of R and B on the blocks of the partitions: = [2, 1] [1, 2]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {5, 9, 10, 11, 12}
Rank2
R,B [7, 7, 7, 6, 3, A, B, C, C, 2, 4, 9], [6, 8, 8, 7, A, 3, A, B, B, C, 1, 5]
π2 [0, 0, 0, 168, 0, 78, 0, 0, 208, 0, 152, 0, 152, 0, 0, 208, 0, 0, 81, 0, 165, 52, 94, 0, 170, 0, 0, 127, 0, 163, 0, 78, 0, 0, 168, 0, 156, 0, 76, 0, 164, 0, 0, 104, 0, 102, 0, 0, 162, 0, 188, 102, 240, 0, 312, 0, 0, 236, 0, 104, 94, 0, 104, 304, 0, 336]
u2 [0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1] (dim 1)
wpp [7, 7, 7, 5, 5, 7, 5, 7, 7, 5, 7, 5]

 

 
1004 . Coloring, {6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = - 6s 2 + 5s 3 - 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, 2, 1, 5]
B: [6, 8, 8, 7, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y2, y1, y2 - y4, 0, y2, y4, y3, 0, 0, y5, y4, y4]

  p' = - s 5 + s 8   p = s 3 - s 6   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y3, y2, y3 + y6, y4, y6, y5, y7]

  p = s 5 - s 8   p' = - s 5 + s 8


 
1005 . Coloring, {6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, C, 4, 5]
B: [6, 8, 8, 7, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y7, y8, y1, y4, y2, 0, 0, y3, y5, y6]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1 + y2 - y7 - y5 - y6 + y4 + y8, y1, y3, 0, 0, y2, y3, y7, y5, y6, y4, y8]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9   p = - s 2 + s 10


 
1006 . Coloring, {6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = 9s 3 + 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, C, 1, 9]
B: [6, 8, 8, 7, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1 + y4, 0, y1, 0, 0, y4, y2, 0, -2 y1 + y4 + y2 - y5 + y3, y5, y4, y3]

  p' = s 5 - s 6   p = s 5 - s 7   p' = - s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 5 y3, -5 y3 + 5 y1 - 5 y2 + 5 y5 + 5 y6 - 5 y8, 6 y1 - 5 y4 + 6 y5 + 6 y6 - 5 y7, 5 y1, 5 y2, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p' = - s 4 - s 6 + s 7 + s 9   p = - s 4 - s 6 + s 7 + s 9


 
1007 . Coloring, {6, 7, 8, 10, 11}

R: [7, 7, 7, 6, A, 3, A, B, B, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y7, y4, y5, 0, 0, y6, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1 - y3 - y4 + y5, 0, y2, 0, 0, y1, y2, y6, y3, y4, y6, y5]

  p = s 6 - s 7   p' = - s 6 + s 7   p' = - s 6 + s 8


 
1008 . Coloring, {6, 7, 8, 10, 12}

R: [7, 7, 7, 6, A, 3, A, B, B, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y5, y3, y4, 0, 0, y2, y1, 0, 2 y2, y6, 2 y4, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 11 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y7 - 5 y8, 5 y1, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9


 
1009 . Coloring, {6, 7, 8, 11, 12}

R: [7, 7, 7, 6, A, 3, A, B, B, C, 4, 9]
B: [6, 8, 8, 7, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, -y2 + y1 + y3 - y5 - y4 + y6 + y7, y2, 0, y1, y3, 0, y5, y4, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y2, y3, y2 - y3 + y1 - y5 + y6 + y4 + y9 - y7 - y8, 0, y1, y5, y6, y4, 0, y9, y7, y8]

  p = s 7 - s 8 + s 9 - s 10


 
1010 . Coloring, {6, 7, 9, 10, 11}

R: [7, 7, 7, 6, A, 3, A, C, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, y5, 2 y4 - 2 y3, y4, y3, y2, 0, 0, y1, 0, 2 y4 - 2 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y5 - y2 - y3 - y4 + y6 + y7, 0, y1, 0, 0, y5, y1, y2, y3, y4, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
1011 . Coloring, {6, 7, 9, 10, 12}

R: [7, 7, 7, 6, A, 3, A, C, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[2 y2, -y1 - 3 y2 - y3 - y4 + 6 y5, y1, 0, 0, y2, y3, 0, y5, y4, 0, y5]

  p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, 0, y9, y8, y6, y7, y5, y4, 0, y3, y1, y2]


 
1012 . Coloring, {6, 7, 9, 11, 12}

R: [7, 7, 7, 6, A, 3, A, C, C, C, 4, 9]
B: [6, 8, 8, 7, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y2, y3, 0, y2 + y3 - y1 + y4 + y5 - y6, y1, 0, y4, y5, 0, y6]

  p = - s 6 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - 3 y4 - y5 - y6 + y7, y1, y2, 0, 2 y4, y3, y4, y5, 0, y6, y7, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
1013 . Coloring, {6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, C, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y5, y3, y4, 0, y1, y2, 0, y8, y9, y7, y6]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y2 + y6 - y3 - y4 - y8 + y5 + y7, 0, y1, 0, y2, y6, y3, y4, 0, y8, y5, y7]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1014 . Coloring, {6, 8, 9, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y4, y3, y5, y6, 0, 0, y9, y8, y7]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, y2, y3, y4, y5, y8, y6, y7]

  p = - s 3 + s 9


 
1015 . Coloring, {6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-6 y1 - 9 y4 - 3 y2 + 13 y3 - 3 y5 + 13 y6, 3 y1 + 3 y4, 3 y1, 0, 0, 3 y4, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 3 + s 9   p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y2, y1, y3, y7, y4, y5, 0, y6, y7, y8]

  p = - s 5 + s 9


 
1016 . Coloring, {6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, C, 4, 9]
B: [6, 8, 8, 7, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -3 y1 - 3 y2 - 3 y3 + 10 y6 - 3 y5, 3 y1, 0, 3 y2, 3 y3, 0, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p' = s 2 - s 7   p = - s 2 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y4, y5, y8, y6, 0, y7, y8, y9]

  p = - s 6 + s 10


 
1017 . Coloring, {6, 8, 10, 11, 12}

R: [7, 7, 7, 6, A, 3, B, B, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y6, y5, 0, 2 y4, y4, y7, 0]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y5, 0, y1, 0, y2, y3, y5, y4, 0, y6, 0, y7]

  p = - s 4 + s 8


 
1018 . Coloring, {6, 9, 10, 11, 12}

R: [7, 7, 7, 6, A, 3, B, C, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -y1 - y2 - y3 - y4 + 6 y7 - y5 - y6, y1, y2, 0, y3, y4, 0, y7, y5, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
1019 . Coloring, {7, 8, 9, 10, 11}

R: [7, 7, 7, 6, A, A, A, B, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y4, y4, y3, y2, 0, 0, y6, y5, y5]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y8, 0, 0, y7, y6, y5, y4, 0, y3, y2]


 
1020 . Coloring, {7, 8, 9, 10, 12}

R: [7, 7, 7, 6, A, A, A, B, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y2 - 3 y6 + 13 y5 - 3 y4 + 13 y3, 0, 0, 0, 3 y2, 3 y6, 0, 3 y5, 3 y4, 3 y2, 3 y3]

  p = - s 3 - s 4 + s 6 + s 7   p' = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 5 y1 + 5 y2 + 5 y4 - 5 y6, 6 y1 + 6 y2 + 6 y4 - 5 y3 - 5 y5, 5 y1, 5 y2, 5 y3, 5 y4, 0, 0, 5 y5, 5 y6]

  p = - s 2 - s 4 + s 5 + s 7   p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8


 
1021 . Coloring, {7, 8, 9, 11, 12}

R: [7, 7, 7, 6, A, A, A, B, C, C, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, 0, y4, 3 y1, 0, y3, y2, y1, y5 - y4 + y3 + y2 - 4 y1]

  p = - s 5 + s 7   p = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y3 - 3 y4 + y5 + y6 - y7, 2 y4, y1, 0, y2, y3, y4, y5, 0, 0, y6, y7]

  p = - s 5 + s 6 - s 7 + s 8   p = - s 5 + s 9


 
1022 . Coloring, {7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 5s 3 - 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, B, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, y3, y4, 0, y5, y7, y6, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y8, 0, 0, y6, y7]


 
1023 . Coloring, {7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, A, A, C, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -y1 + 2 y3, 0, y1, 0, -y2 + 2 y3, y2, 0, y3, 2 y3, 0, y3]

  p = - s 3 + s 5   p = - s 3 + s 7   p = - s 3 + s 6   p = - s 3 + s 4

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y2, 0, 0, y1, 2 y3]

  p = - s 3 + s 8


 
1024 . Coloring, {8, 9, 10, 11, 12}

R: [7, 7, 7, 6, A, A, B, B, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 + 5 y3 - 3 y4 + 8 y5, 0, -3 y2 + 8 y3 - 3 y6 + 5 y5, 0, 3 y1, 3 y2, 0, 3 y3, 3 y6, 3 y4, 3 y5]

  p = - s + s 7   p' = - s + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y4, 0, y1 + y2 + y3 - y6, 0, y1, y2, y5, y3, 0, y4, y5, y6]

  p = - s 4 + s 5 - s 6 + s 7   p = - s 4 + s 8   p' = - s 4 + s 8


 
1025 . Coloring, {2, 3, 4, 5, 6, 7}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, C, 1, 5]
B: [6, 7, 7, 6, A, A, B, B, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[-y4 + y5 + y3, 0, y5, 0, y4, 0, y3, y2, 0, y5 + y3 - y2, y5 + y3 - y1, y1]

  p = - s 5 + s 7   p = - s 5 + s 8   p = - s 5 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 + 8 y3 - 3 y5 + 5 y6, 0, -3 y2 + 5 y3 - 3 y4 + 8 y6, 0, 3 y1, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p' = s - s 7   p = s - s 7


 
1026 . Coloring, {2, 3, 4, 5, 6, 8}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, 3, B, B, B, C, 1, 5]
B: [6, 7, 7, 6, A, A, A, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 3 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -y2 + 2 y3, 0, y2, 0, -y1 + 2 y3, y1, 0, y3, 2 y3, 0, y3]

  p = - s 3 + s 4   p = - s 3 + s 5   p = - s 3 + s 6   p = - s 3 + s 7


 
1027 . Coloring, {2, 3, 4, 5, 6, 9}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7   p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, B, C, C, C, 1, 5]
B: [6, 7, 7, 6, A, A, A, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[y1, 0, y2, 0, y3, 0, y1, 5 y1 - y2 - y3 - y4, 0, 0, y1, y4 ]

  p' = - s 2 + s 6   p = s - s 5   p' = s - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y4, y5, 0, y3, y6, y7, 0]


 
1028 . Coloring, {2, 3, 4, 5, 6, 10}

Ωp(Δ)=0:     p = 9s 3 - 16s 5 - 24s 6 - 16s 7 + 64s 8   p' = 3s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, C, B, 2, 1, 5]
B: [6, 7, 7, 6, A, A, A, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[7 y4, 7 y5, 7 y3, 0, 7 y2, 0, 7 y1, 9 y4 - 7 y5 - 7 y3 - 7 y2 + 9 y1 + 9 y7 - 7 y6, 0, 0, 7 y7, 7 y6]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - y2 - y3 + 3 y4 + y5, 0, y1, 2 y4, 0, y2, y3, y4, y5]

  p = s 5 - s 6   p' = - s 5 + s 6


 
1029 . Coloring, {2, 3, 4, 5, 6, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, B, C, B, C, 4, 5]
B: [6, 7, 7, 6, A, A, A, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 9 y1 - 7 y2 + 9 y3 - 7 y4 + 9 y5 - 7 y6, 7 y1, 7 y2, 0, 7 y3, 7 y4, 0, 0, 7 y5, 7 y6]

  p = - s - s 2 - s 3 + s 5 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y6, 3 y7, 0, 0, 0, 3 y3, 3 y4, 0, 3 y5, -3 y6 - 3 y7 - 3 y3 - 3 y4 + 13 y5 - 3 y2 + 13 y1, 3 y2, 3 y1]

  p = - s 4 - s 5 + s 7 + s 8


 
1030 . Coloring, {2, 3, 4, 5, 6, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, B, C, B, C, 1, 9]
B: [6, 7, 7, 6, A, A, A, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y2, y3, y4, 0, y7, y6]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, y2, y3, y4, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1031 . Coloring, {2, 3, 4, 5, 7, 8}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, C, 1, 5]
B: [6, 7, 7, 6, A, 3, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y4, 0, y3, 0, y2, y7, 0, y6, y5, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -y1 - y2 - y3 - y4 + 6 y7 - y5 - y6, y1, y2, 0, y3, y4, 0, y7, y5, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
1032 . Coloring, {2, 3, 4, 5, 7, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, C, 1, 5]
B: [6, 7, 7, 6, A, 3, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y1, 0, y2, 0, y3, 0, y7, y4, 0, y5, 0, y6]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y4, y2, 0, y3, y5, 0, 2 y6, y6, y7, 0]

  p = s 3 - s 8


 
1033 . Coloring, {2, 3, 4, 5, 7, 10}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, A, C, B, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y4, 0, y5, y9, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -3 y1 - 3 y2 - 3 y3 - 3 y5 + 10 y6, 3 y1, 0, 3 y2, 3 y3, 0, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p = - s 2 + s 7   p' = - s 2 + s 7


 
1034 . Coloring, {2, 3, 4, 5, 7, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, A, A, C, B, C, 4, 5]
B: [6, 7, 7, 6, A, 3, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y6 - 3 y7 + 13 y8 - 3 y5 - 3 y4 + 13 y3, 3 y1, 3 y2, 0, 0, 3 y6, 3 y7, 0, 3 y8, 3 y5, 3 y4, 3 y3]

  p = - s 3 - s 4 + s 8 + s 9


 
1035 . Coloring, {2, 3, 4, 5, 7, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, A, C, B, C, 1, 9]
B: [6, 7, 7, 6, A, 3, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y6, y7, y5, 0, 0, y4, y8, y9]


 
1036 . Coloring, {2, 3, 4, 5, 8, 9}

Ωp(Δ)=0:     p = - s 2 - 4s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8   p = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, A, B, B, C, C, 1, 5]
B: [6, 7, 7, 6, A, 3, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, 0, y7, 0, y6, 0, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y6, 0, y5, y3, 0, y4, y7, y8, y9]


 
1037 . Coloring, {2, 3, 4, 5, 8, 10}

Ωp(Δ)=0:     p = s 3 - 16s 5 - 8s 6 + 16s 7 + 64s 8   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, A, B, B, B, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 2 y2 - 3 y5, 0, 2 y5, 0, y3, y4, 0, y5, y6, 0]

  p' = s 4 - s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y4 + y5 - y2 - y3 + y6, y1, 0, y4, y5, 0, y2, y3, 0, y6]

  p = - s 6 + s 7


 
1038 . Coloring, {2, 3, 4, 5, 8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, B, B, B, C, 4, 5]
B: [6, 7, 7, 6, A, 3, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-y1 - y2 - y4 - y3 + 6 y6 - y5, y1, y2, 0, 0, y4, y3, 0, y6, y5, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1039 . Coloring, {2, 3, 4, 5, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, B, B, C, 1, 9]
B: [6, 7, 7, 6, A, 3, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, 0, y2, y6, y3, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y6, y4, y4, y3, 0, 0, y5, 0, y6]

  p = s 4 - s 7   p' = - s 4 + s 7


 
1040 . Coloring, {2, 3, 4, 5, 9, 10}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 8s 6 - 16s 7   p' = s 3 - 8s 5 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, A, B, C, C, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {1, 7, 11}}    order:   12

See Matrix
 

[2 y4, 2 y3, 5 y4 - 2 y3 - 2 y5, 0, 2 y2, 0, 2 y4, 2 y1, 0, 5 y4 - 2 y2 - 2 y1, 2 y4, 2 y5]

  p' = - s 4 + s 8   p' = - s 3 + s 7   p' = - s 3 + s 4 - s 5 + s 6   p = s 3 - s 4 + s 5 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y3 + y2 - y4 - y5 + y6 + y7, y1, 0, y3, y2, 0, y4, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1041 . Coloring, {2, 3, 4, 5, 9, 11}

Ωp(Δ)=0:     p = - s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, C, C, C, 4, 5]
B: [6, 7, 7, 6, A, 3, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 8 8 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 5 y3 - 2 y4, 2 y3, 5 y3 - 2 y1 - 2 y2, 0, 2 y3, 2 y1, 0, 2 y2, 2 y3, 2 y4]

  p = - s 2 + s 3 - s 4 + s 5   p = - s 2 + s 6   p = - s 2 - s 4 + s 5 + s 7   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y2, y1, 0, 0, y6, y7, 0, y4, y5, y8, 0]


 
1042 . Coloring, {2, 3, 4, 5, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, C, C, C, 1, 9]
B: [6, 7, 7, 6, A, 3, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y2, 0, 5 y2 - 2 y3, 0, 0, 0, 2 y2, -2 y1 + 2 y3, 2 y1, 5 y2 - 2 y3, 2 y2, 2 y3]

  p = - s 3 + s 5   p = - s 3 + s 6   p = - s 3 + s 7   p = - s 3 + s 8   p = - s 3 + s 4

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, y1, y2, y7, y3, y4, 0, 0, y6, y7, 0]

  p = - s 5 + s 8


 
1043 . Coloring, {2, 3, 4, 5, 10, 11}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 8s 6 - 16s 7 + 64s 8   p = 9s 2 + 2s 4 - 16s 6 + 16s 7 - 96s 8   p = - 27s 2 + 8s 5 + 48s 6 - 64s 7 + 256s 8

R: [7, 8, 8, 7, 3, A, B, C, B, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 7 y6, 7 y5, 7 y4, 7 y3, 0, 7 y2, 7 y1, 0, -7 y6 - 7 y5 + 9 y4 - 7 y3 + 9 y2 - 7 y1 + 9 y8 - 7 y7, 7 y8, 7 y7]

  p = s 3 + s 4 + s 5 - s 7 - s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y6, 0, y5, 0, 0, y4, y3, 0, y2, y1, y6 + y5 - y4 - y3 + y2 + y1 - y7, y7]

  p = s 7 - s 8


 
1044 . Coloring, {2, 3, 4, 5, 10, 12}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 + 8s 5 + 32s 7   p = 9s 2 + 2s 4 - 16s 5 - 80s 7 - 32s 8   p = - 27s 2 + 56s 5 + 16s 6 + 256s 7 + 128s 8

R: [7, 8, 8, 7, 3, A, B, C, B, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y6, 0, 0, 0, y3, y5, y4, y6, y8, y7]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y5, y4, 0, 0, y6, y7, y8]


 
1045 . Coloring, {2, 3, 4, 5, 11, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 - 24s 5 + 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, A, B, C, B, C, 4, 9]
B: [6, 7, 7, 6, A, 3, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, 0, y2, y7, y3, y4, y5, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y1, y2, 0, y6, y5, y4, 0, 0, y3, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1046 . Coloring, {2, 3, 4, 6, 7, 8}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, C, 1, 5]
B: [6, 7, 7, 6, 3, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y6, 0, y2, 0, y3, y4, 0, y5, y8, y7]

Omega Rank for B :  cycles: {{9, 12}, {2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -y1 - y2 - y3 - y4 + 6 y7 - y5 - y6, y1, y2, 0, y3, y4, 0, y7, y5, y6, y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
1047 . Coloring, {2, 3, 4, 6, 7, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, C, 1, 5]
B: [6, 7, 7, 6, 3, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, y3, 0, y2, 0, -2 y3 + 2 y1, y1, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, y4, y5, 0, y3, y2, 0, 2 y4, y6, y7, 0]

  p = s 2 - s 8


 
1048 . Coloring, {2, 3, 4, 6, 7, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, A, 3, A, C, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, y7, 0, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y4, 0, y5, y2, 0, y3, y8, y6, y7]


 
1049 . Coloring, {2, 3, 4, 6, 7, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, A, C, B, C, 4, 5]
B: [6, 7, 7, 6, 3, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y6, y3, y1, 0, y2, y3, 0, y5, y6, y4]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y4, 2 y5, 2 y6, 0, 0, 2 y7, 2 y3, 0, 2 y1, -2 y4 + 11 y5 + 11 y6 + 11 y7 - 2 y3 - 39 y1 + 11 y2, 2 y2, 3 y5 + 3 y6 + 3 y7 - 11 y1 + 3 y2]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
1050 . Coloring, {2, 3, 4, 6, 7, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, A, C, B, C, 1, 9]
B: [6, 7, 7, 6, 3, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y2, y3, y1, y4, y5, y6, 0, 0, y9, y7, y8]


 
1051 . Coloring, {2, 3, 4, 6, 8, 9}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, B, B, C, C, 1, 5]
B: [6, 7, 7, 6, 3, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[y4, 0, y3, 0, y2, 0, y1, 4 y4 + 4 y3 + 5 y2 - y1 - 6 y5, 0, -2 y4 - 2 y3 - 4 y2 + 5 y5, y5, 5 y4 + 5 y3 + 6 y2 - 8 y5]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y8, y2, 0, y3, y4, 0, y5, y6, y7, y8]

  p = - s 6 + s 9


 
1052 . Coloring, {2, 3, 4, 6, 8, 10}

Ωp(Δ)=0:     p' = s 3 - 8s 6 - 16s 7   p = s 3 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, 3, B, B, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 4 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, y3, y2, 0, 2 y2, 0, y1, y7, 0, y6, y5, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, -2 y1 + 2 y2, y2, 0, -5 y1 + 3 y2 - y3 + y4, y3, 0, y4]

  p' = - s 4 + s 5   p = s 4 - s 5   p' = - s 4 + s 6


 
1053 . Coloring, {2, 3, 4, 6, 8, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, B, B, B, C, 4, 5]
B: [6, 7, 7, 6, 3, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 5 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -2 y1 - 2 y4 + 9 y2 + 9 y3 - 11 y5, 2 y1, 2 y4 - 2 y2 - 2 y3 + 2 y5, 0, 2 y2, 2 y3, 0, -2 y4 + 7 y2 + 7 y3 - 9 y5, 2 y4, 2 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[2 y1, -3 y1 - y2 - y3 + 6 y5 - y4, y1, 0, 0, y2, y3, 0, y5, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
1054 . Coloring, {2, 3, 4, 6, 8, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, B, B, B, C, 1, 9]
B: [6, 7, 7, 6, 3, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, 0, y6, y4, y5, y3, y2, y4]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y6, y5, y5, y3, 0, 0, y4, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1055 . Coloring, {2, 3, 4, 6, 9, 10}

Ωp(Δ)=0:     p = 3s 3 + s 4 + 4s 5 + 8s 7 - 16s 8   p = - 9s 3 - 8s 5 + 8s 6 - 16s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, B, C, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {1, 7, 11}}   

See Matrix
 

[y5, 5 y5 - y1 - y2 - y3 - y4 - y6, y1, 0, y2, 0, y5, y3, 0, y4, y5, y6]

  p = s 2 - s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1056 . Coloring, {2, 3, 4, 6, 9, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, B, C, C, C, 4, 5]
B: [6, 7, 7, 6, 3, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 5 y4 - y2 - y1 - y5 - y3, y4, y2, 0, y4, y1, 0, y5, y4, y3]

  p' = s 3 - s 6   p' = - s 4 + s 7   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, 0, y4, y5, 0, 2 y3, y6, y7, 0]

  p = - s 5 + s 8


 
1057 . Coloring, {2, 3, 4, 6, 9, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, B, C, C, C, 1, 9]
B: [6, 7, 7, 6, 3, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y3, 0, 5 y3 - 2 y2 - 2 y1, 0, 0, 0, 2 y3, 2 y2, 2 y1, 5 y3 - 2 y2 - 2 y1, 2 y3, 2 y2 + 2 y1]

  p' = s 4 - s 5   p' = s 3 - s 5   p = s 3 - s 6   p' = - s 5 + s 7   p' = - s 5 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, y1, y2, -2 y1 + 2 y2, y4, y5, 0, 0, y6, -2 y1 + 2 y2, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1058 . Coloring, {2, 3, 4, 6, 10, 11}

Ωp(Δ)=0:     p = s 2 + 8s 5 + 16s 6   p' = s 2 + 8s 5 + 16s 6   p' = s 3 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, 3, B, C, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 5 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, -7 y1 + 9 y2 - 7 y3 + 9 y8 - 7 y4 - 7 y5 + 9 y6 - 7 y7, 7 y1, 7 y2, 7 y3, 0, 7 y8, 7 y4, 0, 7 y5, 7 y6, 7 y7]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, y2 + y3 - y5, y1, 0, y2, y3, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8


 
1059 . Coloring, {2, 3, 4, 6, 10, 12}

Ωp(Δ)=0:     p' = s 3 - 8s 5 - 8s 6 - 16s 7   p' = s 2 - 8s 4 - 8s 5 - 16s 6   p = s 2 - 8s 5 - 80s 6 - 64s 7 - 128s 8

R: [7, 8, 8, 7, A, 3, B, C, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y6, 0, 0, 0, y5, y3, y4, y6, y7, y8]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y5, y3, y4, 0, 0, y8, y6, y7]


 
1060 . Coloring, {2, 3, 4, 6, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7   p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, B, C, B, C, 4, 9]
B: [6, 7, 7, 6, 3, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y4, 0, 0, y3, y2, y1, y5, y7, y6]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3 - y6, 0, y2, y3, y4, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = s 4 - s 7   p' = s 5 - s 8


 
1061 . Coloring, {2, 3, 4, 7, 8, 9}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, C, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y6, 0, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y7, y6, y5, 0, y4, y3, 0, y2, 0, y1, y7]

  p = s 3 - s 8


 
1062 . Coloring, {2, 3, 4, 7, 8, 10}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, A, A, B, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 2 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5 + y6, y1, 0, 0, y2, 0, y3, y4, 0, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, y2, y2, 0, y2, y2, 0, 3 y2 - y1, 0, y2, y1]

  p' = s 4 - s 6   p' = s 3 - s 5   p' = s 2 - s 6   p' = s - s 5   p = s - s 7


 
1063 . Coloring, {2, 3, 4, 7, 8, 11}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, A, A, B, B, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, 0, y5, y6, 0, y4, y3, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[y6, y5, y4, 0, 0, y3, y2, 0, y1, 0, -y6 - y5 - y4 - y3 - y2 + 6 y1, y1]

  p' = - s 2 + s 7   p = - s 2 + s 7


 
1064 . Coloring, {2, 3, 4, 7, 8, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, B, B, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y1, y2, y3, y7, y4, 0, 0, 0, y5, y6]

  p = - s 3 + s 8


 
1065 . Coloring, {2, 3, 4, 7, 9, 10}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 8, 8, 7, A, A, A, C, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, y5, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, 0, y5, 0, y6, y7]


 
1066 . Coloring, {2, 3, 4, 7, 9, 11}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, C, C, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, y5, y1, 0, y4, y5, 0, y2, 0, y3]

  p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y4, y2, 0, 0, y3, y5, 0, y4, 0, y6, 0]

  p = - s 2 + s 7


 
1067 . Coloring, {2, 3, 4, 7, 9, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, A, A, C, C, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y2, 0, 0, 0, 0, 0, y1, y2, y3, y4, 0, y5]

  p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y3, y1, y2, y3, y4, y6, 0, 0, 0, y5, 0]

  p = s 2 - s 7


 
1068 . Coloring, {2, 3, 4, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-5 y2 - 5 y1 - 5 y5 + 11 y6 - 5 y3 + 11 y4, 0, 5 y2, 0, 0, 5 y1, 5 y5, 0, 5 y6, 0, 5 y3, 5 y4]

  p = - s - s 2 + s 6 + s 7


 
1069 . Coloring, {2, 3, 4, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y8, y7, 0, 0, 0, 0, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y3, y2, y1, y7, y6, 0, 0, 0, y5, y4]


 
1070 . Coloring, {2, 3, 4, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, y3, y4, y5, 0, 0, 0, y6, y7]

  p = - s 3 + s 8


 
1071 . Coloring, {2, 3, 4, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 24s 6 - 16s 7 - 64s 8   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, A, B, B, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, -y7 + y6 + y5 - y4 - y3 + y2 + y1, y7, 0, y6, y5, 0, y4, y3, y2, y1]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1072 . Coloring, {2, 3, 4, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, B, B, C, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y4, 2 y3, 0, 2 y2, 7 y4 - 27 y3 - 2 y2 + 16 y1, 0, 3 y4 - 7 y3 + 4 y1, 2 y1, 4 y4 - 16 y3 + 10 y1]

  p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y5, y6, y4, 0, 0, y3, y2, 0, y1, y7, y9, y8]


 
1073 . Coloring, {2, 3, 4, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, B, B, C, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y1, 0, 0, 0, 0, 0, 2 y2, 7 y1 + 7 y2 - 9 y3 + 7 y4 - 9 y5, 2 y3, 7 y1 + 7 y2 - 9 y3 + 7 y4 - 9 y5, 2 y4, 2 y5]

  p' = - s 2 - s 3 + s 5 + s 6   p = - s 2 + s 4 + s 5 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, y4, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1074 . Coloring, {2, 3, 4, 8, 10, 11}

Ωp(Δ)=0:     p = s 2 - 8s 5 - 16s 6   p' = s 2 - 8s 5 - 16s 6   p' = s 3 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, A, B, B, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 7 vs 7 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, 0, y4, y5, 0, y6, y7, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-y1 + y2 + y3 - y4 - y5 + y6, 0, y1, 0, 0, y2, y3, 0, y4, y5, 0, y6]

  p = - s 6 + s 7


 
1075 . Coloring, {2, 3, 4, 8, 10, 12}

Ωp(Δ)=0:     p' = s 2 + 8s 4 + 8s 5 + 16s 6   p' = s 3 + 8s 5 + 8s 6 + 16s 7   p = s 2 + 8s 5 - 48s 6 - 64s 7 - 128s 8

R: [7, 8, 8, 7, A, A, B, B, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, y4, y5, y5, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, 0, y7]


 
1076 . Coloring, {2, 3, 4, 8, 11, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 8, 8, 7, A, A, B, B, B, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y4, y3, y2, y3, y6, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y6, 0, y4, y4, y5, 0, 0, y3, 0, y2]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
1077 . Coloring, {2, 3, 4, 9, 10, 11}

Ωp(Δ)=0:     p' = s 7   p' = s 6   p' = s 5   p = s 2   p' = s 2   p' = s 3   p' = s 4

R: [7, 8, 8, 7, A, A, B, C, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 8 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, y1, 0, y1, y1, 0, y1, y1, 0, y1, y1, y1]

  p = - s + s 2   p = - s + s 3   p = - s + s 4   p = - s + s 8   p = - s + s 5   p = - s + s 6   p = - s + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y1, 0, 0, y1, y1, 0, y1, y1, y1, y1]

  p' = s 2 - s 7   p' = s 5 - s 7   p' = s 3 - s 7   p' = s 6 - s 7   p' = s 4 - s 7   p' = s - s 7   p = s - s 8


` See 8-level graph

`

M              \ ;       N

$ [ [0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1] , [0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1] , [1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1] , [0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1] , [0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1] , [1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2] , [0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1] , [1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0] ] $     $ [ [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0] ] $

τ= 20 , r'= 7/8

R: [7, 8, 8, 7, A, A, B, C, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, B, B, C, 1, 9]

Ranges

Action of R on ranges, [[2], [2]]
Action of B on ranges, [[1], [1]]

Cycles:    R , {{2, 5, 8, 10, 12}, {4, 7, 11}},   B , {{1, 3, 6, 7, 9, 10, 11, 12}}

β({1, 3, 6, 7, 9, 10, 11, 12}) = 1/2
β({2, 4, 5, 7, 8, 10, 11, 12}) = 1/2

Partitions
α([{10}, {11}, {2, 3}, {12}, {5, 6}, {1, 4}, {8, 9}, {7}]) = 1/1

b1 = {10} ` , ` b2 = {11} ` , ` b3 = {2, 3} ` , ` b4 = {12} ` , ` b5 = {5, 6} ` , ` b6 = {1, 4} ` , ` b7 = {8, 9} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [5, 8, 1, 7, 4, 2, 3, 6] [8, 7, 5, 1, 6, 2, 4, 3]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-8 partition graph.

`

Right Group
Coloring {2, 3, 4, 9, 10, 11}
Rank8
R,B [7, 8, 8, 7, A, A, B, C, C, 2, 4, 5], [6, 7, 7, 6, 3, 3, A, B, B, C, 1, 9]
π2 [0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2]
u2 [1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 1)
wpp [2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1]
π8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

 

 
1078 . Coloring, {2, 3, 4, 9, 10, 12}

Ωp(Δ)=0:     p = 9s 2 + 2s 4 - 16s 6 + 16s 7 - 96s 8   p = - 27s 2 + 8s 5 + 48s 6 - 64s 7 + 256s 8   p = - 6s 2 + s 3 + 8s 6 - 16s 7 + 64s 8

R: [7, 8, 8, 7, A, A, B, C, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y4, 5 y4 - y1 - y2 - y3 - y5, 0, 0, 0, 0, y4, y1, y2, y3, y4, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6   p = - s 4 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y5, y3, y4, y2, y1, 0, 0, y8, y6, y7]


 
1079 . Coloring, {2, 3, 4, 9, 11, 12}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7   p = 9s 2 - 28s 4 + 40s 5 - 16s 6 - 96s 7 + 64s 8

R: [7, 8, 8, 7, A, A, B, C, C, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 3 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, 0, y2, y1, -2 y1 + 5 y2 - y3, y1, y2, y3]

  p' = - s 2 + s 6   p' = - s 2 + s 4   p = - s 2 + s 4   p = - s 2 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y7, y4, y5, 0, 0, y6, y7, 0]

  p = - s 5 + s 8


 
1080 . Coloring, {2, 3, 4, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 8s 6 - 16s 7   p' = s 3 - 8s 5 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, A, B, C, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, 0, y8, y7, y6, y5, y4, y3]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y4, 0, y3, 0, y2, y1, y8, 0, 0, y7, y6, y5]


 
1081 . Coloring, {2, 3, 5, 6, 7, 8}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, C, 1, 5]
B: [6, 7, 7, 7, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 9 vs 9 4 vs 8

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y8, 0, y9, 0, y7, y5, y6, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -y1 + 2 y4, 0, y1, 0, -y2 + 2 y4, y2, 0, y4, -y3 + 2 y4, y3, y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 8   p = - s 4 + s 7


 
1082 . Coloring, {2, 3, 5, 6, 7, 9}

Ωp(Δ)=0:     p = - 2s 2 + s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, C, 1, 5]
B: [6, 7, 7, 7, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y3, 0, y1, 0, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 4 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, y4, y3, 0, 2 y4, y5, y6, 0]

  p = s 4 - s 7


 
1083 . Coloring, {2, 3, 5, 6, 7, 10}

R: [7, 8, 8, 6, 3, 3, A, C, B, 2, 1, 5]
B: [6, 7, 7, 7, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y9, y8, y7, 0, y6, y5, y4, y3, 0, y2, y5, y1]

  p = - s 6 + s 10

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y5, 0, y4, y3, 0, y2, y1, -y5 - y3 + 2 y2 + 2 y1, -y4 + y2 + y1]

  p' = - s 3 + s 6   p = - s 3 + s 6


 
1084 . Coloring, {2, 3, 5, 6, 7, 11}

R: [7, 8, 8, 6, 3, 3, A, C, B, C, 4, 5]
B: [6, 7, 7, 7, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y6, y5, y4, y3, y2, y1, 0, y5, y2, -y6 + y4 + y3 + y1]

  p' = - s 4 + s 8   p' = - s 4 + s 5 - s 6 + s 7   p = s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y6, 3 y5, 0, 0, 0, 3 y4, 3 y3, 0, 3 y2, -3 y6 - 3 y3 + 8 y2 + 5 y1, -3 y5 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = - s + s 7   p = - s + s 7


 
1085 . Coloring, {2, 3, 5, 6, 7, 12}

R: [7, 8, 8, 6, 3, 3, A, C, B, C, 1, 9]
B: [6, 7, 7, 7, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y9, 0, 0, y8, y7, y6, y5, y4, y3, y2]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, 0, y3, y2, y6, y1, 0, 0, y5, y7, y6]

  p = s 5 - s 8


 
1086 . Coloring, {2, 3, 5, 6, 8, 9}

R: [7, 8, 8, 6, 3, 3, B, B, C, C, 1, 5]
B: [6, 7, 7, 7, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, 0, y7, 3 y4]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y7, 0, y6, 0, y5, y4, 0, y3, y2, y1, y5]

  p = s 5 - s 8


 
1087 . Coloring, {2, 3, 5, 6, 8, 10}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, 2, 1, 5]
B: [6, 7, 7, 7, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 4 vs 6

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y3, y2, 0, 2 y3, y3, y4, y5, 0, 0, y6, 0]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y4, 0, y4, y3, 0, y2, y1, 0, y4 - y3 + y2 + y1]

  p' = s 4 - s 5   p = s 4 - s 6

 ยป SYNC'D !RANK'D


 
1088 . Coloring, {2, 3, 5, 6, 8, 11}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, C, 4, 5]
B: [6, 7, 7, 7, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 3 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y2, y3, y4, y1, y5, y6, 0, 0, y7, 2 y5]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[y1, -y1 + 2 y3, 0, 0, 0, y2, -y2 + 2 y3, 0, y3, 2 y3, 0, y3]

  p' = s 5 - s 6   p' = s 4 - s 6   p' = s 3 - s 6   p = s 3 - s 7


 
1089 . Coloring, {2, 3, 5, 6, 8, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, C, 1, 9]
B: [6, 7, 7, 7, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y3, y5, y4, 2 y2 - 2 y3, 0, y6, 2 y3]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y3, 0, 2 y5, y4, y5, y2, 0, 0, y1, 0, 2 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1090 . Coloring, {2, 3, 5, 6, 9, 10}

Ωp(Δ)=0:     p = - 2s 2 + 5s 3 + 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, 2, 1, 5]
B: [6, 7, 7, 7, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[5 y7, 10 y4, 5 y6, 0, 5 y5, 5 y4, 5 y3, 5 y2, 0, 0, 5 y1, 11 y7 - 5 y6 - 5 y5 - 15 y4 + 11 y3 - 5 y2 + 11 y1]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8   p = s 2 - s 5 - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, 0, y3, y4, 0, y1, y2, y5 - y3 - y4 + y1 + y2 - y6, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1091 . Coloring, {2, 3, 5, 6, 9, 11}

Ωp(Δ)=0:     p = 2s 2 + 5s 3 - 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, C, 4, 5]
B: [6, 7, 7, 7, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 8 vs 8 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y8, y4, y5, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y4, y3, 0, 0, 0, y2, y1, 0, y7, y6, y5, 0]


 
1092 . Coloring, {2, 3, 5, 6, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, C, 1, 9]
B: [6, 7, 7, 7, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y7, 0, 5 y6, 0, 0, 5 y5, 5 y4, 5 y2, 5 y3, 0, 5 y1, 11 y7 - 5 y6 - 5 y5 + 11 y4 - 5 y2 - 5 y3 + 11 y1]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, y2, 2 y3, y3, y4, 0, 0, y5, 2 y3, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6

 ยป SYNC'D !RANK'D


 
1093 . Coloring, {2, 3, 5, 6, 10, 11}

R: [7, 8, 8, 6, 3, 3, B, C, B, 2, 4, 5]
B: [6, 7, 7, 7, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 2 y4, -y1 + y2 + y3 - 3 y4 + y5 + y6 - y7, y1, y2, y3, y4, y5, 0, 0, y6, y7]

  p = - s 5 + s 6 - s 7 + s 8   p = - s 5 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1 - y2 - y3 + 4 y4 + y5, 0, 0, 0, 0, y1, 3 y4, 0, y2, y3, y4, y5]

  p = - s 5 + s 7   p = - s 5 + s 6


 
1094 . Coloring, {2, 3, 5, 6, 10, 12}

R: [7, 8, 8, 6, 3, 3, B, C, B, 2, 1, 9]
B: [6, 7, 7, 7, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y8, 2 y6, y7, 0, 0, y6, y4, y5, y3, 0, y2, y1]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y5, y3, 0, 0, y4, y5, y6]

  p = - s 4 + s 7


 
1095 . Coloring, {2, 3, 5, 6, 11, 12}

R: [7, 8, 8, 6, 3, 3, B, C, B, C, 4, 9]
B: [6, 7, 7, 7, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y5, y4, 0, y1, y2, y3, y6, 0, y7, y8]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, 0, 0, y2, y3, y4, 0, 0, y5, y6, y6]

  p' = - s 4 + s 7   p = s 4 - s 7


 
1096 . Coloring, {2, 3, 5, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, C, 1, 5]
B: [6, 7, 7, 7, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 9 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y6, y5, 0, y8, y9, y7]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1, -y5 + y1, y3, 0, y5, y2, 0, y1, y5, y4, y5]

  p' = s 3 - s 6   p' = s 5 - s 8   p' = s 4 - s 7   p = s 3 - s 9


 
1097 . Coloring, {2, 3, 5, 7, 8, 10}

R: [7, 8, 8, 6, 3, A, A, B, B, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 4 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y1, y3, 0, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 3 y4, 3 y3, 0, 3 y2, 3 y4 + 3 y3, 0, 3 y1, 3 y2, -3 y2 + 3 y4 + 3 y3, -3 y2 + 7 y4 + 7 y3 - 3 y1]

  p' = s 5 - s 7   p' = s 4 - s 6   p' = s 3 - s 7   p = s 3 - s 7


 
1098 . Coloring, {2, 3, 5, 7, 8, 11}

R: [7, 8, 8, 6, 3, A, A, B, B, C, 4, 5]
B: [6, 7, 7, 7, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y7, y3, y4, y5, 0, y6, y8, y9]

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[y1, y2, -y1 - y2 - y3 - y4 + 6 y5 - y6 - y7, 0, 0, y3, y4, 0, y5, y6, y7, y5]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
1099 . Coloring, {2, 3, 5, 7, 8, 12}

R: [7, 8, 8, 6, 3, A, A, B, B, C, 1, 9]
B: [6, 7, 7, 7, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, y5, y4, 2 y5, y3, y2, 0, 0, y1, y7, 2 y3]

  p' = s 5 - s 8   p = s 5 - s 8


 
1100 . Coloring, {2, 3, 5, 7, 9, 10}

R: [7, 8, 8, 6, 3, A, A, C, C, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[2 y5, y8, y7, 0, y6, y5, y4, y3, 0, y2, 0, y1]

  p = s 5 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -y4 + y6, y1, 0, y4, y2, 0, y3, y4, y5, y6]

  p = s 4 - s 7   p' = s 4 - s 7


 
1101 . Coloring, {2, 3, 5, 7, 9, 11}

R: [7, 8, 8, 6, 3, A, A, C, C, C, 4, 5]
B: [6, 7, 7, 7, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y3, 2 y7, y2, y1, y7, y6, 0, y5, 0, y4]

  p = s 4 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y3, y1, y2, 0, 0, y7, y6, 0, 2 y5, y5, y4, 0]

  p = - s 3 + s 8


 
1102 . Coloring, {2, 3, 5, 7, 9, 12}

R: [7, 8, 8, 6, 3, A, A, C, C, C, 1, 9]
B: [6, 7, 7, 7, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, y1, 0, 0, y1, -3 y1 + 2 y2, y2, y5, y4, 0, y3]

  p = - s 4 + s 6   p' = - s 4 + s 6   p = - s 4 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y1, y3, 4 y1 - 2 y5, 2 y1 - y5, y4, 0, 0, y5, y6, 0 ]

  p = s 4 - s 7   p' = s 4 - s 7


 
1103 . Coloring, {2, 3, 5, 7, 10, 11}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y6, y5, y4, y3, y2, y8, y1, 0, y9, y8, y7]

  p = - s 6 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y3, 0, 3 y2, 0, 0, 3 y1, -3 y3 - 3 y2 - 3 y1 + 10 y6 + 10 y5 - 3 y4, 0, 3 y6, 3 y5, 3 y4, 3 y6 + 3 y5]

  p' = s 2 - s 7   p = s 2 - s 7


 
1104 . Coloring, {2, 3, 5, 7, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y3, y4, y5, y6, y9, y7, y8]

  p = - s 2 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y1, y2, -2 y1 - 2 y2 + 5 y3 + 5 y4 - 4 y5, y3, 4 y1 + 4 y2 - 6 y3 - 6 y4 + 5 y5, 0, 0, 5 y1 + 5 y2 - 8 y3 - 8 y4 + 6 y5, y4, y5]

  p' = - s 4 + s 7   p = - s 3 + s 6   p' = - s 3 + s 6


 
1105 . Coloring, {2, 3, 5, 7, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, C, 4, 9]
B: [6, 7, 7, 7, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y6, y2, y3, y7, y4, y8, y5]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y9, y8, y7, 0, y6, y5, y4, 0, 0, y3, y2, y1]


 
1106 . Coloring, {2, 3, 5, 8, 9, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, -y6 + y5 + 3 y4, y4, y3, y2, 0, -y6 + y5 + 2 y4, y1, y4]

  p = s 5 - s 8   p' = s 6 - s 9   p' = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, -y1 + y7 + y6 - y5 - y4 + y3 + y2, 0, y7, y6, 0, y5, y4, y3, y2]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1107 . Coloring, {2, 3, 5, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, C, 4, 5]
B: [6, 7, 7, 7, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1 - y2 + y8 - y7 - y6 - y5 + y4 + y3, y2, y8, y7, y6, 0, y5, y4, y3]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y9, y8, 0, 0, y6, y7, 0, y5, y4, y3, y2]


 
1108 . Coloring, {2, 3, 5, 8, 9, 12}

Ωp(Δ)=0:     p = - 2s 2 + s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, C, 1, 9]
B: [6, 7, 7, 7, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 8 vs 10 4 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y2 - 3 y4 + 3 y3, 0, 3 y4, 0, 0, 3 y4, 3 y2, 3 y3, 3 y1, -3 y4 + 3 y3, 3 y2 + 3 y3, 7 y2 - 3 y1 + 4 y3]

  p' = s 4 - s 6   p' = s 5 - s 7   p' = s 3 - s 7   p = s 3 - s 7   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y2, y1, y1 + y5, y1 + y5, y5, y4, 0, 0, y3, y5, y5]

  p' = - s 5 + s 8   p = s 3 - s 6   p' = s 4 - s 7   p' = s 3 - s 6

 ยป SYNC'D !RANK'D


 
1109 . Coloring, {2, 3, 5, 8, 10, 11}

R: [7, 8, 8, 6, 3, A, B, B, B, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y2 - y3 + 3 y4 + y5 + y6 - y7, y1, y2, 2 y4, y3, y4, y5, 0, y6, y7, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-y1 + y4 + y5 - y6 - y3 + y2, 0, y1, 0, 0, y4, y5, 0, y6, y3, 0, y2]

  p = - s 6 + s 7


 
1110 . Coloring, {2, 3, 5, 8, 10, 12}

R: [7, 8, 8, 6, 3, A, B, B, B, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, y7, y5, 0, 0, y5, y3, y4, 2 y5, y1, y2, 0]

  p' = s 5 - s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y3, y2, y3, y4, 0, 0, y5, 0, y6]

  p = - s 4 + s 7


 
1111 . Coloring, {2, 3, 5, 8, 11, 12}

R: [7, 8, 8, 6, 3, A, B, B, B, C, 4, 9]
B: [6, 7, 7, 7, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, y5, y7, y4, y2, y3, y1, y6 - y5 + y4 + y2 + y3 - y1]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y3 - 2 y1, y4, y5, 0, y3, y1, y2, 0, 0, y6, 0, 2 y3 - 2 y1]

  p = s 4 - s 7   p' = s 4 - s 7


 
1112 . Coloring, {2, 3, 5, 9, 10, 11}

R: [7, 8, 8, 6, 3, A, B, C, C, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y5 + y6 + y7 - y8 - y9, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

  p = s 7 - s 8 + s 9 - s 10

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1113 . Coloring, {2, 3, 5, 9, 10, 12}

R: [7, 8, 8, 6, 3, A, B, C, C, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[5 y1, 11 y1 - 10 y2 + 11 y3 - 5 y4 - 5 y7 - 5 y5 + 11 y6 - 5 y8, 5 y2, 0, 0, 5 y2, 5 y3, 5 y4, 5 y7, 5 y5, 5 y6, 5 y8]

  p = - s 5 - s 6 + s 8 + s 9   p = s 5 - s 7 - s 8 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y1, y2, y3, y4, 0, 0, y5, 2 y3, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
1114 . Coloring, {2, 3, 5, 9, 11, 12}

R: [7, 8, 8, 6, 3, A, B, C, C, C, 4, 9]
B: [6, 7, 7, 7, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y6, y4, 0, y5, y6, y3, y1, y2, y3, y4 - y5 + y1 + y2]

  p = - s 6 + s 7   p = - s 6 + s 8   p = - s 6 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y7, y4, y5, 0, 0, y6, y7, 0]

  p = s 5 - s 8


 
1115 . Coloring, {2, 3, 5, 10, 11, 12}

Ωp(Δ)=0:     p = 6s 2 - 5s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, C, B, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y8, y7, y6, 0, y5, y7, y4, y3, y2, y1, -y8 + y6 - y5 + y4 + y3 + y2 - y1]

  p' = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9   p = s 2 - s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y7, y6, y8]


 
1116 . Coloring, {2, 3, 6, 7, 8, 9}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, C, 1, 5]
B: [6, 7, 7, 7, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y6, y2, 0, y6, y3, 0, y6 + y4, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1117 . Coloring, {2, 3, 6, 7, 8, 10}

Ωp(Δ)=0:     p = s 2 + 3s 3 - 6s 4 + 8s 5 - 24s 6 + 32s 7 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y4 + y3 + y1 + y2 - y7 - y6 + y5, y4, y3, 0, 2 y1, y1, y2, y7, 0, y6, y5, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, y1, -y2 + y3 + y4 - y5 + y6, 0, y1, y2, 0, y3, y4, y5, y6]

  p = s 3 + s 4 - s 6 - s 7   p = - s 3 + s 5 + s 6 - s 8


M              \ ;       N

$ [ [0, 352, 168, 0, 402, 0, 270, 0, 0, 0, 540, 0] , [352, 0, 0, 0, 0, 84, 0, 624, 0, 672, 0, 0] , [168, 0, 0, 0, 0, 336, 0, 210, 0, 576, 0, 442] , [0, 0, 0, 0, 0, 270, 0, 0, 402, 168, 0, 892] , [402, 0, 0, 0, 0, 446, 0, 221, 0, 663, 0, 0] , [0, 84, 336, 270, 446, 0, 260, 0, 0, 0, 336, 0] , [270, 0, 0, 0, 0, 260, 0, 260, 667, 681, 0, 1326] , [0, 624, 210, 0, 221, 0, 260, 0, 0, 0, 417, 0] , [0, 0, 0, 402, 0, 0, 667, 0, 0, 0, 663, 0] , [0, 672, 576, 168, 663, 0, 681, 0, 0, 0, 704, 0] , [540, 0, 0, 0, 0, 336, 0, 417, 663, 704, 0, 804] , [0, 0, 442, 892, 0, 0, 1326, 0, 0, 0, 804, 0] ] $     $ [ [0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1] , [0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1] , [0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0] , [0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0] , [0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1] , [0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0] ] $

τ= 72 , r'= 1/2

R: [7, 8, 8, 6, A, 3, A, B, B, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, B, C, C, C, 4, 9]

Ranges

Action of R on ranges, [[22], [22], [24], [24], [4], [10], [26], [7], [10], [26], [7], [18], [9], [21], [6], [17], [11], [28], [8], [11], [2], [28], [28], [8], [19], [5], [5], [1], [3]]
Action of B on ranges, [[20], [20], [9], [21], [13], [24], [25], [25], [24], [25], [25], [23], [24], [25], [25], [23], [11], [12], [12], [28], [15], [29], [29], [29], [27], [16], [16], [16], [14]]

Cycles:    R , {{1, 2, 7, 8, 10, 11}},   B , {{4, 7, 11}, {9, 12}}

β({1, 2}) = 11/433
β({1, 3}) = 21/1732
β({1, 5}) = 201/6928
β({1, 7}) = 135/6928
β({1, 11}) = 135/3464
β({2, 6}) = 21/3464
β({2, 8}) = 39/866
β({2, 10}) = 21/433
β({3, 6}) = 21/866
β({3, 8}) = 105/6928
β({3, 10}) = 18/433
β({3, 12}) = 221/6928
β({4, 6}) = 135/6928
β({4, 9}) = 201/6928
β({4, 10}) = 21/1732
β({4, 12}) = 223/3464
β({5, 6}) = 223/6928
β({5, 8}) = 221/13856
β({5, 10}) = 663/13856
β({6, 7}) = 65/3464
β({6, 11}) = 21/866
β({7, 8}) = 65/3464
β({7, 9}) = 667/13856
β({7, 10}) = 681/13856
β({7, 12}) = 663/6928
β({8, 11}) = 417/13856
β({9, 11}) = 663/13856
β({10, 11}) = 22/433
β({11, 12}) = 201/3464

Partitions
α([{2, 3, 4, 5, 7, 11}, {1, 6, 8, 9, 10, 12}]) = 1/1

b1 = {2, 3, 4, 5, 7, 11} ` , ` b2 = {1, 6, 8, 9, 10, 12}

Action of R and B on the blocks of the partitions: = [2, 1] [1, 2]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {2, 3, 6, 7, 8, 10}
Rank2
R,B [7, 8, 8, 6, A, 3, A, B, B, 2, 1, 5], [6, 7, 7, 7, 3, A, B, C, C, C, 4, 9]
π2 [352, 168, 0, 402, 0, 270, 0, 0, 0, 540, 0, 0, 0, 0, 84, 0, 624, 0, 672, 0, 0, 0, 0, 336, 0, 210, 0, 576, 0, 442, 0, 270, 0, 0, 402, 168, 0, 892, 446, 0, 221, 0, 663, 0, 0, 260, 0, 0, 0, 336, 0, 260, 667, 681, 0, 1326, 0, 0, 417, 0, 0, 663, 0, 704, 0, 804]
u2 [1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1] (dim 1)
wpp [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]

 

 
1118 . Coloring, {2, 3, 6, 7, 8, 11}

Ωp(Δ)=0:     p = s 2 - 3s 3 - 6s 4 - 8s 5 - 24s 6 - 32s 7 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, B, C, 4, 5]
B: [6, 7, 7, 7, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 0, -y1 + y2 - y3 + y4 - y5 + y6 - y7 + y8, y1, y2, y3, y4, y5, 0, y6, y7, y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-y3 + 3 y6 - y4, -y1 - y2 + 3 y6 - y5, y1, 0, 0, y2, y3, 0, y6, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = - s 2 + s 8   p' = - s 2 + s 8


M              \ ;       N

$ [ [0, 152, 52, 0, 0, 78, 0, 0, 168, 0, 156, 0] , [152, 0, 0, 0, 0, 0, 208, 0, 0, 81, 0, 165] , [52, 0, 0, 0, 94, 0, 170, 0, 0, 127, 0, 163] , [0, 0, 0, 0, 168, 0, 78, 0, 0, 208, 0, 152] , [0, 0, 94, 168, 0, 76, 0, 164, 0, 0, 104, 0] , [78, 0, 0, 0, 76, 0, 102, 0, 0, 162, 0, 188] , [0, 208, 170, 78, 0, 102, 0, 102, 240, 0, 312, 0] , [0, 0, 0, 0, 164, 0, 102, 0, 0, 236, 0, 104] , [168, 0, 0, 0, 0, 0, 240, 0, 0, 94, 0, 104] , [0, 81, 127, 208, 0, 162, 0, 236, 94, 0, 304, 0] , [156, 0, 0, 0, 104, 0, 312, 0, 0, 304, 0, 336] , [0, 165, 163, 152, 0, 188, 0, 104, 104, 0, 336, 0] ] $     $ [ [0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0] ] $

τ= 74 , r'= 1/2

R: [7, 8, 8, 6, A, 3, A, B, B, C, 4, 5]
B: [6, 7, 7, 7, 3, A, B, C, C, 2, 1, 9]

Ranges

Action of R on ranges, [[23], [23], [10], [25], [14], [26], [27], [18], [26], [26], [27], [18], [21], [21], [22], [17], [11], [30], [15], [11], [12], [9], [30], [30], [15], [31], [19], [31], [19], [16], [13]]
Action of B on ranges, [[20], [20], [21], [22], [3], [25], [6], [24], [10], [25], [6], [24], [10], [25], [6], [24], [11], [12], [2], [30], [7], [28], [31], [31], [5], [8], [29], [8], [29], [1], [4]]

Cycles:    R , {{5, 10, 12}, {3, 4, 6, 8, 11}},   B , {{1, 2, 6, 7, 10, 11}, {9, 12}}

β({1, 2}) = 19/606
β({1, 3}) = 13/1212
β({1, 6}) = 13/808
β({1, 9}) = 7/202
β({1, 11}) = 13/404
β({2, 7}) = 13/303
β({2, 10}) = 27/1616
β({2, 12}) = 55/1616
β({3, 5}) = 47/2424
β({3, 7}) = 85/2424
β({3, 10}) = 127/4848
β({3, 12}) = 163/4848
β({4, 5}) = 7/202
β({4, 7}) = 13/808
β({4, 10}) = 13/303
β({4, 12}) = 19/606
β({5, 6}) = 19/1212
β({5, 8}) = 41/1212
β({5, 11}) = 13/606
β({6, 7}) = 17/808
β({6, 10}) = 27/808
β({6, 12}) = 47/1212
β({7, 8}) = 17/808
β({7, 9}) = 5/101
β({7, 11}) = 13/202
β({8, 10}) = 59/1212
β({8, 12}) = 13/606
β({9, 10}) = 47/2424
β({9, 12}) = 13/606
β({10, 11}) = 19/303
β({11, 12}) = 7/101

Partitions
α([{1, 5, 7, 10, 12}, {2, 3, 4, 6, 8, 9, 11}]) = 1/1

b1 = {1, 5, 7, 10, 12} ` , ` b2 = {2, 3, 4, 6, 8, 9, 11}

Action of R and B on the blocks of the partitions: = [1, 2] [2, 1]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {2, 3, 6, 7, 8, 11}
Rank2
R,B [7, 8, 8, 6, A, 3, A, B, B, C, 4, 5], [6, 7, 7, 7, 3, A, B, C, C, 2, 1, 9]
π2 [152, 52, 0, 0, 78, 0, 0, 168, 0, 156, 0, 0, 0, 0, 0, 208, 0, 0, 81, 0, 165, 0, 94, 0, 170, 0, 0, 127, 0, 163, 168, 0, 78, 0, 0, 208, 0, 152, 76, 0, 164, 0, 0, 104, 0, 102, 0, 0, 162, 0, 188, 102, 240, 0, 312, 0, 0, 236, 0, 104, 94, 0, 104, 304, 0, 336]
u2 [1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1] (dim 1)
wpp [5, 7, 7, 7, 5, 7, 5, 7, 7, 5, 7, 5]

 

 
1119 . Coloring, {2, 3, 6, 7, 8, 12}

R: [7, 8, 8, 6, A, 3, A, B, B, C, 1, 9]
B: [6, 7, 7, 7, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 2 y1 - 3 y3, y2, 2 y5, y3, y4, 0, 0, y5, y6, 2 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1120 . Coloring, {2, 3, 6, 7, 9, 10}

R: [7, 8, 8, 6, A, 3, A, C, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[4 y4 - 2 y2, y5, y4, 0, y1, 2 y4 - y2, y2, y3, 0, y6, 0, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y2, y6, 0, y7, y3, y4, y5]

  p = - s 5 + s 8


 
1121 . Coloring, {2, 3, 6, 7, 9, 11}

R: [7, 8, 8, 6, A, 3, A, C, C, C, 4, 5]
B: [6, 7, 7, 7, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y7, 2 y4, y6, y5, y4, y3, 0, y2, 0, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 - y3 + y2 - y4 - y5 + y6, y1, y3, 0, 0, y2, y4, 0, 2 y3, y5, y6, 0]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1122 . Coloring, {2, 3, 6, 7, 9, 12}

R: [7, 8, 8, 6, A, 3, A, C, C, C, 1, 9]
B: [6, 7, 7, 7, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, y5, 0, 0, y4, 2 y5 - y4, y3, y2, -y5 + 2 y3, 0, y1]

  p' = s 5 - s 7   p' = s 4 - s 6   p = s 4 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, -y3 + 2 y5, y2, 2 y3, y3, y4, 0, 0, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1123 . Coloring, {2, 3, 6, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y5, y4, y3, y2, y8, y6, 0, y9, y8, y7]

  p = - s 5 + s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 + y2 - y3 - y4 - y7 + y5 + y6, 0, y1, 0, 0, y2, y3, 0, y4, y7, y5, y6]

  p = - s 7 + s 8

 ยป SYNC'D !RANK'D


 
1124 . Coloring, {2, 3, 6, 7, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, y10, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, 0, 0, y3, y2, y1]


 
1125 . Coloring, {2, 3, 6, 7, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, C, 4, 9]
B: [6, 7, 7, 7, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y4, y5, 0, y1, y2, y3, y7, y8, y9, y6]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 + y4 - y5 - y6 + y7 + y8, y1, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1126 . Coloring, {2, 3, 6, 8, 9, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, y3, y2, 0, y2 + y6, y6, y1, y8, 0, y7, y5, y6]

  p' = - s 6 + s 9   p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2 - y3 - y4 + y5 + y6, 0, y1, y2, 0, y3, y4, y5, y6]

  p' = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = s 2 - s 8


 
1127 . Coloring, {2, 3, 6, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, C, 4, 5]
B: [6, 7, 7, 7, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -3 y6 + 5 y5 - 3 y2 - 3 y3 - 3 y4 + 5 y1 - 3 y7 + 5 y8, 3 y6, 3 y5, 3 y2, 3 y3, 3 y4, 0, 3 y1, 3 y7, 3 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y4, y5, 0, 0, y3, y2, 0, y8, y7, y6, y5]

  p = - s 6 + s 9


 
1128 . Coloring, {2, 3, 6, 8, 9, 12}

Ωp(Δ)=0:     p = 2s 2 - s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, C, 1, 9]
B: [6, 7, 7, 7, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, 0, -3 y1 + 7 y6 - 3 y5 - 3 y4 + 10 y3 - 3 y2, 0, 0, 3 y6, 3 y5, 3 y4, 3 y3, 3 y6, 3 y2, 3 y6 + 3 y3]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y1, y3, y3, y6, y5, 0, 0, y4, y6, y6]

  p' = - s 4 + s 7   p = s 4 - s 7   p' = - s 5 + s 8


 
1129 . Coloring, {2, 3, 6, 8, 10, 11}

R: [7, 8, 8, 6, A, 3, B, B, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 4 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y8, y7, y6, 2 y4, y5, y4, y3, 0, y2, y1, 0]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, y4, 0, 0, -5 y4 + 2 y1, y1, 0, -8 y4 + 3 y1 - y3 + y2, y3, 0, y2]

  p' = s 4 - s 6   p' = s 5 - s 6   p = s 4 - s 7


 
1130 . Coloring, {2, 3, 6, 8, 10, 12}

R: [7, 8, 8, 6, A, 3, B, B, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2 + y5, y2, 0, 0, y5, y3, y4, 2 y5, y5, y6, 0]

  p' = - s 4 + s 7   p' = - s 5 + s 8   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y6, 2 y4, y5, y4, y3, 0, 0, y2, 0, y1]

  p = - s 2 + s 7


 
1131 . Coloring, {2, 3, 6, 8, 11, 12}

R: [7, 8, 8, 6, A, 3, B, B, B, C, 4, 9]
B: [6, 7, 7, 7, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y2, y3, 0, y1, y8, y6, y7, y8, y4, y5]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y3 - 2 y4, y1, y2, 0, y3, y4, y5, 0, 0, y6, 0, 2 y3 - 2 y4]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
1132 . Coloring, {2, 3, 6, 9, 10, 11}

R: [7, 8, 8, 6, A, 3, B, C, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[0, y1, y5, y6, y7, y2, y3, y4, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y2 + y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y2, y3, 0, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1133 . Coloring, {2, 3, 6, 9, 10, 12}

R: [7, 8, 8, 6, A, 3, B, C, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y1, 5 y7 + 5 y3, 5 y7, 0, 0, 5 y3, 5 y4, 5 y5, 5 y6, 5 y3, 5 y2, 11 y1 - 10 y7 - 15 y3 + 11 y4 - 5 y5 - 5 y6 + 11 y2]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9   p = - s 4 + s 10

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, y6, 2 y4, y7]

  p = - s 3 + s 8


 
1134 . Coloring, {2, 3, 6, 9, 11, 12}

R: [7, 8, 8, 6, A, 3, B, C, C, C, 4, 9]
B: [6, 7, 7, 7, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y6, y7, 0, y3, y4, y5, y1, y4, y2, -y6 - y7 + y3 + y5 + y1 + y2]

  p = s 7 - s 8   p' = - s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, y4, 0, 2 y6 - 2 y4, y3, y2, 0, 0, y1, 2 y6 - 2 y4, 0]

  p' = s 4 - s 7   p = s 4 - s 7


 
1135 . Coloring, {2, 3, 6, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, C, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 9 vs 10 8 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y1, y4, y2, 0, y3, y7, y5, y6, y7, y8, y9]

  p = - s 3 + s 10

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y8, 0, y7, 0, y6, y5, y4, 0, 0, y3, y2, y1]


 
1136 . Coloring, {2, 3, 7, 8, 9, 10}

R: [7, 8, 8, 6, A, A, A, B, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 5 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, y3, y6, y4, y5, 0, y7, y8, y6]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, y2, y1, 0, y5, y4, 0, 3 y2 - 4 y5, 0, y3, 3 y5]

  p = s 3 - s 6   p' = s 3 - s 6


 
1137 . Coloring, {2, 3, 7, 8, 9, 11}

R: [7, 8, 8, 6, A, A, A, B, C, C, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 2 y6, y2, 0, 0, y3, y4, 0, y5, 0, y7, y6]

  p = - s 3 + s 8


 
1138 . Coloring, {2, 3, 7, 8, 9, 12}

R: [7, 8, 8, 6, A, A, A, B, C, C, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, 2 y2, y4, y5, y6, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y5, y3, y2, y1, y5, y4, 0, 0, 0, y6, y5]

  p = s 4 - s 7   p' = s 4 - s 7


 
1139 . Coloring, {2, 3, 7, 8, 10, 11}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, 0, y4, 2 y1, -y3 + y4 - 3 y1 + y2 + y5 - y6, y1, y2, 0, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-3 y1 - 3 y5 - 3 y4 + 5 y3 - 3 y2 + 5 y6, 0, 3 y1, 0, 0, 3 y5, 3 y4, 0, 3 y3, 0, 3 y2, 3 y6]

  p = - s - s 2 + s 6 + s 7

 ยป SYNC'D !RANK'D


 
1140 . Coloring, {2, 3, 7, 8, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y3, 0, 0, 0, y1, y6, y7, 2 y1, y4, y5, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y6, 0, 0, 0, y5, 4 y4]

  p = - s 4 + s 7

 ยป SYNC'D !RANK'D


 
1141 . Coloring, {2, 3, 7, 8, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, y2, y1, 2 y1, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y7, y2, 0, y3, y4, y5, 0, 0, 0, y6, y7]

  p = - s 3 + s 8


 
1142 . Coloring, {2, 3, 7, 9, 10, 11}

Ωp(Δ)=0:     p = - 2s 2 - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, 2 y6, y4, y5, y6, y2, 0, y3, 0, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, 0, y3, 0, 0, y1, y6, 0, y7, 0, y4, y5]


 
1143 . Coloring, {2, 3, 7, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y1, 0, 0, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, 0, y6, 2 y4]

  p = - s 4 + s 7


 
1144 . Coloring, {2, 3, 7, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y2, 0, y1, y2, 2 y2, y3, y4, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y3, y2, 0, y3, y4, y5, 0, 0, 0, y6, 0]

  p = - s 2 + s 7

 ยป SYNC'D !RANK'D


 
1145 . Coloring, {2, 3, 7, 10, 11, 12}

R: [7, 8, 8, 6, A, A, A, C, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y4, 0, y3, 0, y2, y1, y6, y5, y9, y8, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, 0, 0, 0, y2, y1]


 
1146 . Coloring, {2, 3, 8, 9, 10, 11}

R: [7, 8, 8, 6, A, A, B, B, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y2 - y3 + 2 y7 + y4 + y5 - y6, 0, y1, y2, y3, y7, y4, 0, y5, y6, y7]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y1 - y5 + y4, 0, 0, y1 + y4 - y2, y1 + y4 - y3, 0, y5, y4, y3, y2]

  p' = - s + s 2 - s 5 + s 6   p' = - s + s 3 - s 5 + s 7   p = s - s 2 + s 5 - s 6


 
1147 . Coloring, {2, 3, 8, 9, 10, 12}

R: [7, 8, 8, 6, A, A, B, B, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y6, y4, 0, 0, y5, y6, y7]

  p = - s 3 + s 8


 
1148 . Coloring, {2, 3, 8, 9, 11, 12}

R: [7, 8, 8, 6, A, A, B, B, C, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, 2 y3, y6, y5, y4, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, y4, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1149 . Coloring, {2, 3, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, B, B, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, y5, 2 y4, y7, y6, 0]

  p = s 2 - s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y4, y5, y3, 0, 0, y6, 0, y7]


 
1150 . Coloring, {2, 3, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, A, B, C, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y4, 0, y3, 0, y1, y2, y8, y7, y5, y6, y9]

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y5, 0, y4, 0, y3, y2, y1, 0, 0, y6, y7, y8]


 
1151 . Coloring, {2, 4, 5, 6, 7, 8}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, C, 1, 5]
B: [6, 7, 8, 6, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y6, 0, 3 y5, 0, 3 y4, 3 y3, 3 y2, 3 y1, 8 y6 - 3 y5 + 8 y4 - 3 y3 - 11 y2 - 11 y1 + 8 y7, 3 y7, 5 y6 + 5 y4 - 8 y2 - 8 y1 + 5 y7]

  p' = s 2 - s 8   p = s 2 - s 8


 
1152 . Coloring, {2, 4, 5, 6, 7, 9}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, C, 1, 5]
B: [6, 7, 8, 6, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y6, 0, y1, 0, y3, 0, y2, y6, 0, y5, 0, y4]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y5 - y1 + y2 + 3 y3 + y4 - y6, 0, y5, 0, y1, y2, y3, 2 y3, y4, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8


 
1153 . Coloring, {2, 4, 5, 6, 7, 10}

R: [7, 8, 7, 7, 3, 3, A, C, B, 2, 1, 5]
B: [6, 7, 8, 6, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y4, 0, y5, y9, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6 - 2 y5 - y4 - y3 + y2 + y1, 0, y6, y5, y5, y4, y3, y2, y1]

  p = s 6 - s 7   p' = - s 6 + s 7


 
1154 . Coloring, {2, 4, 5, 6, 7, 11}

R: [7, 8, 7, 7, 3, 3, A, C, B, C, 4, 5]
B: [6, 7, 8, 6, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y6, 0, y5, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 + 5 y4 - 3 y5 + 8 y7, -3 y3 + 8 y4 - 3 y6 + 5 y7, 0, 0, 0, 3 y3, 3 y1, 3 y2, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
1155 . Coloring, {2, 4, 5, 6, 7, 12}

R: [7, 8, 7, 7, 3, 3, A, C, B, C, 1, 9]
B: [6, 7, 8, 6, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 2 y3, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y4, y5, y7, 0, y8, y6, y7]

  p = - s 3 + s 9


 
1156 . Coloring, {2, 4, 5, 6, 8, 9}

R: [7, 8, 7, 7, 3, 3, B, B, C, C, 1, 5]
B: [6, 7, 8, 6, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y6, 0, y5, 0, y4, 0, y3, y2, 0, 0, y1, 3 y2]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, 0, y2, 0, y3, y4, y5, y6, y7, y8, y9]


 
1157 . Coloring, {2, 4, 5, 6, 8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, 2, 1, 5]
B: [6, 7, 8, 6, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 2 y2 - 2 y4, y2, 0, 2 y2 - 2 y4, 0, y3, y4, 0, 0, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y3, 0, y1, y3, y3, y4, y2, 0, y5]

  p = s 4 - s 6   p' = s 4 - s 6


 
1158 . Coloring, {2, 4, 5, 6, 8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, C, 4, 5]
B: [6, 7, 8, 6, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, 0, y6, 2 y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[10 y1 + 10 y2 + 10 y3 - 22 y4 + 10 y5 - 22 y6, y1, 0, 0, 0, y2, y3, 5 y1 + 5 y2 + 5 y3 - 11 y4 + 5 y5 - 11 y6, y4, y5, 0, y6]

  p = s 3 + s 4 - s 6 - s 7   p' = - s 3 - s 4 + s 6 + s 7


 
1159 . Coloring, {2, 4, 5, 6, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, C, 1, 9]
B: [6, 7, 8, 6, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 0, 2 y2, 0, 0, 0, y3, y2, y4, 0, y5, 2 y2]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, 0, 2 y6, y2, -2 y6 + 2 y4, y1, y6, 0, y5, 0, y4]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
1160 . Coloring, {2, 4, 5, 6, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, 2, 1, 5]
B: [6, 7, 8, 6, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y5, 0, y3, 0, y4, y6, 0, 0, y7, y8]

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8


 
1161 . Coloring, {2, 4, 5, 6, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, C, 4, 5]
B: [6, 7, 8, 6, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y3, y2, 0, y6, y7, 0, 0, y4, y5]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, 2 y5, y6, y7, 0]

  p = - s 5 + s 8


 
1162 . Coloring, {2, 4, 5, 6, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, C, 1, 9]
B: [6, 7, 8, 6, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y1, 0, -14 y1 - 14 y4 + 18 y3 - 14 y2 + 18 y5, 0, 0, 0, 5 y4, -7 y1 - 7 y4 + 9 y3 - 7 y2 + 9 y5, 5 y3, 0, 5 y2, 5 y5]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, 0, y4, 2 y3, y1, y2, y3, 0, y7, y6, 0]

  p = s 5 - s 8


 
1163 . Coloring, {2, 4, 5, 6, 10, 11}

R: [7, 8, 7, 7, 3, 3, B, C, B, 2, 4, 5]
B: [6, 7, 8, 6, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, 0, y7, y8, 0, 0, y5, y6]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y7, y7, y3, y4, y5, y6]

  p = - s 6 + s 8


 
1164 . Coloring, {2, 4, 5, 6, 10, 12}

R: [7, 8, 7, 7, 3, 3, B, C, B, 2, 1, 9]
B: [6, 7, 8, 6, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y2, 0, 0, 0, y6, y7, y4, 0, y5, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y2, y1, y3, y3, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1165 . Coloring, {2, 4, 5, 6, 11, 12}

R: [7, 8, 7, 7, 3, 3, B, C, B, C, 4, 9]
B: [6, 7, 8, 6, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 2 y5, y1, 0, 0, y6, y5, y4, 0, y3, y2]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y6 + y7, y4, y3, y7, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1166 . Coloring, {2, 4, 5, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, C, 1, 5]
B: [6, 7, 8, 6, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y8, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y4, y3, y2, 0, y1, y6, y7, y5, y9, y10, y8]


 
1167 . Coloring, {2, 4, 5, 7, 8, 10}

R: [7, 8, 7, 7, 3, A, A, B, B, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y4, 0, y5, y6, 0, y7, y8, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, y3, y2, y4, y1, -y7 - y6 + y5 + y3 + y2 + y1]

  p' = s 7 - s 8   p = s 7 - s 9


 
1168 . Coloring, {2, 4, 5, 7, 8, 11}

R: [7, 8, 7, 7, 3, A, A, B, B, C, 4, 5]
B: [6, 7, 8, 6, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y3, y1, y2, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   10

See Matrix
 

[y2, y3, y1, 0, 0, y4, y5, y6, y7, y8, y9, y10]


 
1169 . Coloring, {2, 4, 5, 7, 8, 12}

R: [7, 8, 7, 7, 3, A, A, B, B, C, 1, 9]
B: [6, 7, 8, 6, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, 0, 0, y2, y3, y7, y5, y6, y4]

  p = s 2 - s 8

Omega Rank for B :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, y1, y3, y2, y7, y6, y5, y4, 0, y9, y8, y10]


 
1170 . Coloring, {2, 4, 5, 7, 9, 10}

R: [7, 8, 7, 7, 3, A, A, C, C, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y4, 0, y5, y6, 0, y7, 0, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y3, y6, y4, y5, y6, y7, y8]

  p = - s 4 + s 9


 
1171 . Coloring, {2, 4, 5, 7, 9, 11}

R: [7, 8, 7, 7, 3, A, A, C, C, C, 4, 5]
B: [6, 7, 8, 6, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, 2 y4, y6, 0, y5, y4, 0, y3, 0, y2]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y7, y7, y8, 0]

  p = - s 4 + s 9


 
1172 . Coloring, {2, 4, 5, 7, 9, 12}

R: [7, 8, 7, 7, 3, A, A, C, C, C, 1, 9]
B: [6, 7, 8, 6, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, y2, 0, 0, 0, y1, y2, y3, y4, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, 0]


 
1173 . Coloring, {2, 4, 5, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, y4, 0, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[-3 y1 - 3 y2 - 3 y4 - 3 y3 - 3 y6 + 10 y5, 0, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, -3 y3 + 3 y5, 3 y3, 3 y6, 3 y5]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8


 
1174 . Coloring, {2, 4, 5, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y2, y1, y7, 0, 0, 0, y6, y3, y4, y5, y9, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -3 y3 + 5 y4 - 3 y1 - 3 y2 - 3 y6 + 5 y5 - 3 y7 + 5 y8, 3 y3, 3 y4, 3 y1, 3 y2, 3 y6, 0, 3 y5, 3 y7, 3 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1175 . Coloring, {2, 4, 5, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, C, 4, 9]
B: [6, 7, 8, 6, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, 0, y5, y7, y4, y3, y2, y1]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]


 
1176 . Coloring, {2, 4, 5, 8, 9, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y4, 0, y2, 0, y3, y4, 0, y6, y5, y6]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y4, y3, 0, y2, y1, y9, y8, y7, y6, y5]


 
1177 . Coloring, {2, 4, 5, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, C, 4, 5]
B: [6, 7, 8, 6, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}, {2, 7, 10}}   

See Matrix
 

[5 y1 - 3 y2 - 3 y3 + 5 y7 - 3 y6 - 3 y5 + 5 y4 - 3 y8 - 3 y9, 3 y1, 3 y2, 0, 0, 3 y3, 3 y7, 3 y6, 3 y5, 3 y4, 3 y8, 3 y9]

  p = - s - s 2 - s 3 + s 8 + s 9 + s 10


 
1178 . Coloring, {2, 4, 5, 8, 9, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, C, 1, 9]
B: [6, 7, 8, 6, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-6 y2 - 3 y1 + 10 y4 - 3 y3, 0, 3 y2, 0, 0, 0, 3 y1, 3 y2, -3 y2 + 3 y4, 3 y2, 3 y3, 3 y4]

  p' = - s 2 + s 5   p' = - s 3 + s 6   p' = - s 4 + s 7   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y10, y9, y8, y7, y6, y5, y4, 0, y3, y2, y1]


 
1179 . Coloring, {2, 4, 5, 8, 10, 11}

R: [7, 8, 7, 7, 3, A, B, B, B, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 2 y2 - 3 y4, y1, 2 y4, 0, y6, y5, 0, y4, y3, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y1, 0, -7 y1 + y2 + y3 + 3 y4 - y5, 0, 0, -4 y1 + 2 y4, y1, y2, y3, y4, 0, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8


 
1180 . Coloring, {2, 4, 5, 8, 10, 12}

R: [7, 8, 7, 7, 3, A, B, B, B, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y5, 0, 0, 0, y3, y6, 2 y5, y5, y4, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y3, 2 y4, y1, y2, y4, y5, 0, y6, 0, y7]

  p = s 5 - s 8


 
1181 . Coloring, {2, 4, 5, 8, 11, 12}

R: [7, 8, 7, 7, 3, A, B, B, B, C, 4, 9]
B: [6, 7, 8, 6, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, 0, y6, y3, y2, y3, y4, y5]

  p = s 4 - s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, 0, y9]


 
1182 . Coloring, {2, 4, 5, 9, 10, 11}

R: [7, 8, 7, 7, 3, A, B, C, C, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, 0, y5, y6, 0, y9, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1183 . Coloring, {2, 4, 5, 9, 10, 12}

R: [7, 8, 7, 7, 3, A, B, C, C, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y4 - y1 + 2 y2 + 2 y3 - y5, y4 + y2 + y3 - y6, y4, 0, 0, 0, y1, y2, y3, y4, y5, y6]

  p' = s 5 - s 8   p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -y1 + y2 - y3 + y4 - y5 + y6 - y7 + y8, y1, y2, y3, y4, y5, 0, y6, y7, y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1184 . Coloring, {2, 4, 5, 9, 11, 12}

R: [7, 8, 7, 7, 3, A, B, C, C, C, 4, 9]
B: [6, 7, 8, 6, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, y2, -y2 - y1 + 2 y4 - y3, 0, 0, y1, y2, -2 y2 + y4, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[y1 - y2 + y5 - y6 + y4 - y3 + y7 - y8, y1, y2, 0, y5, y6, y4, y3, 0, y7, y8, 0]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1185 . Coloring, {2, 4, 5, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, B, C, B, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y6, y1, 0, 0, y3, y4, y5, y6, y8, y7]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[3 y6, 0, 3 y5, 0, 3 y4, 3 y3, 3 y2, 3 y1, 0, 3 y4 + 3 y2, -3 y6 - 3 y5 + 7 y4 - 3 y3 + 7 y2 - 3 y1, 3 y4 + 3 y2]

  p' = s 2 - s 7   p = s 2 - s 7   p' = s 3 - s 8


 
1186 . Coloring, {2, 4, 6, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, C, 1, 5]
B: [6, 7, 8, 6, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, y4, 0, y1, 0, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y3, y1, y2, 0, y9, y10, y8, y6, y7, y5, y4]


 
1187 . Coloring, {2, 4, 6, 7, 8, 10}

R: [7, 8, 7, 7, A, 3, A, B, B, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y3, y2, 0, 2 y2, 0, y6, y7, 0, y4, y5, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y4, y6, 0, y7, -y1 + y4, y1, y2, y3, y4, y5]

  p' = s 6 - s 8   p = - s 6 + s 8


 
1188 . Coloring, {2, 4, 6, 7, 8, 11}

R: [7, 8, 7, 7, A, 3, A, B, B, C, 4, 5]
B: [6, 7, 8, 6, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1, y2, 0, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[3 y3, 3 y2, 3 y8 + 3 y7 - 3 y4, 0, 0, 3 y1, -3 y3 - 3 y2 - 3 y1 + 10 y8 + 10 y7 - 3 y6 - 3 y5, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4]

  p' = s 3 - s 9   p = s 3 - s 9


 
1189 . Coloring, {2, 4, 6, 7, 8, 12}

R: [7, 8, 7, 7, A, 3, A, B, B, C, 1, 9]
B: [6, 7, 8, 6, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, 0, y3, y2, y7, y6, y5, y4]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 3 y1, 3 y2 + 3 y4 - 3 y8, -3 y1 + 10 y2 - 3 y5 - 3 y3 + 10 y4 - 3 y6 - 3 y7, 3 y2, 3 y5, 3 y3, 3 y4, 0, 3 y6, 3 y7, 3 y8]

  p = - s - s 3 + s 7 + s 9   p' = - s - s 3 + s 7 + s 9


 
1190 . Coloring, {2, 4, 6, 7, 9, 10}

R: [7, 8, 7, 7, A, 3, A, C, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y3, y2, y3, 0, y1, 0, y6, y7, 0, y5, 0, y4]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
1191 . Coloring, {2, 4, 6, 7, 9, 11}

R: [7, 8, 7, 7, A, 3, A, C, C, C, 4, 5]
B: [6, 7, 8, 6, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y3, 2 y3, y1, 0, y2, y3, 0, y4, 0, y5]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y3, y7, y8, 0]

  p = - s 3 + s 9


 
1192 . Coloring, {2, 4, 6, 7, 9, 12}

R: [7, 8, 7, 7, A, 3, A, C, C, C, 1, 9]
B: [6, 7, 8, 6, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, y2, 0, 0, 0, y1, y2, y3, y4, 0, y5]

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, y6, y7, y8, 0, y5, y9, 0]


 
1193 . Coloring, {2, 4, 6, 7, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y2, y8, y1, y3, 0, y6, y4, 0, y5, y8, y7]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y3, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p = s 7 - s 9


 
1194 . Coloring, {2, 4, 6, 7, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, 0, y4, y7, y5, y6, y8, y9]

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4, y2, y3, y1, y5, y6, 0, y7, y8, y9]


 
1195 . Coloring, {2, 4, 6, 7, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, C, 4, 9]
B: [6, 7, 8, 6, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y4, y3, 0, 0, y2, y4, y1, y6, y5, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y10, y9]


 
1196 . Coloring, {2, 4, 6, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y8, 0, y3, 0, y5, y4, 0, y6, y7, y8]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1 + 2 y2 - y3 - y4 - y5 + y6 + y7, 0, y1, y2, y3, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
1197 . Coloring, {2, 4, 6, 8, 9, 11}

Ωp(Δ)=0:     p = - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, C, 4, 5]
B: [6, 7, 8, 6, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y1, -7 y1 - 6 y2 + 4 y4 + 5 y3, 5 y1 + 5 y2 - 2 y4 - 4 y3, 0, y2, y1, 0, -8 y1 - 8 y2 + 5 y4 + 6 y3, y4, y3]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p' = s 2 - s 5   p = s 2 - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, 0, y4, y8, y9, y7, y10, y5, y6]


 
1198 . Coloring, {2, 4, 6, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, C, 1, 9]
B: [6, 7, 8, 6, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[3 y4, 0, 3 y3, 0, 0, 0, 3 y2, 3 y3, 3 y1, 3 y3, -3 y4 + 4 y3 - 3 y2 + 10 y1, 3 y3 + 3 y1]

  p' = - s 3 + s 6   p = - s 2 + s 5   p' = - s 2 + s 5   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 11 y6 - 5 y4 + 11 y5 - 5 y3 - 5 y1 + 11 y2 - 5 y8 - 5 y9 + 11 y7, 5 y6, 5 y4, 5 y5, 5 y3, 5 y1, 5 y2, 0, 5 y8, 5 y9, 5 y7]

  p = - s 4 - s 5 - s 6 + s 8 + s 9 + s 10


 
1199 . Coloring, {2, 4, 6, 8, 10, 11}

R: [7, 8, 7, 7, A, 3, B, B, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 4 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, y5, y6, 2 y5, 0, y7, y1, 0, y2, y3, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y3, 0, y3, 0, 0, 2 y2, y3, y2, y1, y2 - y1 + y4, 0, y4]

  p' = s 4 - s 6   p = s 4 - s 7   p' = - s 6 + s 7   p' = s 5 - s 6


 
1200 . Coloring, {2, 4, 6, 8, 10, 12}

R: [7, 8, 7, 7, A, 3, B, B, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 4 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, y4, 0, 0, 0, y6, y5, 2 y4, y4, y3, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, -y1 + y3 - y2, 2 y1, -2 y1 + y3, y2, y1, y3 - y4, 0, y4, 0, y3]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8


 
1201 . Coloring, {2, 4, 6, 8, 11, 12}

R: [7, 8, 7, 7, A, 3, B, B, B, C, 4, 9]
B: [6, 7, 8, 6, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, 0, y2, y4, y3, y4, y5, y6]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[-3 y1 + 10 y2 - 3 y3 - 3 y4 + 10 y5 - 3 y6, 3 y1, 3 y2 + 3 y5 - 3 y7, 0, 3 y2, 3 y3, 3 y4, 3 y5, 0, 3 y6, 0, 3 y7]

  p = - s 3 - s 5 + s 6 + s 8   p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9


 
1202 . Coloring, {2, 4, 6, 9, 10, 11}

Ωp(Δ)=0:     p = s 4 - 16s 8   p' = s 4 - 8s 7   p' = s 5 - 4s 7   p' = s 6 - 2s 7

R: [7, 8, 7, 7, A, 3, B, C, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
4 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, y1 + y2 - y6 + y5 - y3 - y4 + y8 - y7, y1, y2, y6, 0, y5, y3, 0, y4, y8, y7]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[2 y5 + y7 - y6 - y4 - y2 + y3 + y1, 0, y5, 0, 0, y7, y5, y6, y4, y2, y3, y1]

  p = s 3 - s 9   p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1203 . Coloring, {2, 4, 6, 9, 10, 12}

R: [7, 8, 7, 7, A, 3, B, C, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y4 - y1 + 2 y2 + 2 y3 - y5, y4 + y2 + y3 - y6, y4, 0, 0, 0, y1, y2, y3, y4, y5, y6]

  p' = - s 5 + s 8   p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y5 + y6 - y2 - y3, -y1 + y5 + y6, y1, y2, y3, -y4 + y5 + y6, 0, y4, y5, y6]

  p' = s 3 - s 4 + s 7 - s 8   p' = s 2 - s 4 + s 6 - s 8   p = s 2 - s 5 + s 6 - s 9


 
1204 . Coloring, {2, 4, 6, 9, 11, 12}

R: [7, 8, 7, 7, A, 3, B, C, C, C, 4, 9]
B: [6, 7, 8, 6, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, y2, -y2 - y1 - y3 + 2 y4, 0, 0, y1, y2, -2 y2 + y4, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, 0]


 
1205 . Coloring, {2, 4, 6, 10, 11, 12}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, C, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y5, y1, 0, 0, y8, y7, y6, y5, y4, y3]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[-y3 + y4 + y5, 0, y2, 0, y3, y1, y4 + y5 - y2 - y1, y4 + y5 - y6, 0, y6, y4, y5]

  p = - s 2 + s 4 - s 6 + s 8   p = - s 2 + s 5 - s 6 + s 9   p = - s 2 + s 3 - s 6 + s 7


 
1206 . Coloring, {2, 4, 7, 8, 9, 10}

R: [7, 8, 7, 7, A, A, A, B, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y5 - y4 + y6 - y7, y1, 0, 0, y2, 0, y3, y5, 0, y4, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y2, 0, y8, y5, y6, y7, 0, y4, y3]


 
1207 . Coloring, {2, 4, 7, 8, 9, 11}

R: [7, 8, 7, 7, A, A, A, B, C, C, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, 0, y3, y4, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y9, y8, y7, 0, 0, y6, y5, y4, y3, 0, y2, y1]


 
1208 . Coloring, {2, 4, 7, 8, 9, 12}

R: [7, 8, 7, 7, A, A, A, B, C, C, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y3, y2, y7, y6, y4, y5]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y2, y7, y3, y4, y5, y6, 0, 0, y9, y8]


 
1209 . Coloring, {2, 4, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, 0, y4, y5, 0, y6, y7, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y7, 0, 0, y8, y6, y2, y3, 0, y4, y5]


 
1210 . Coloring, {2, 4, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, y4, y5, y6, y7, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y6, y5, y4, y3, y2, y1, 0, 0, y8, y7]


 
1211 . Coloring, {2, 4, 7, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y6, y4, y5, y7, 0, 0, y8, y9]


 
1212 . Coloring, {2, 4, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, 0, y4, y5, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, 0, y7, 2 y4]

  p = s 3 - s 8


 
1213 . Coloring, {2, 4, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, 0, y3, y4, y5, y6, 0, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, 0, y7, 2 y5]

  p = - s 3 + s 8


 
1214 . Coloring, {2, 4, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y3, 0, 0, y5, y3, y4, y2, 0, y1]

  p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y4, y2, 0, y4, y5, y6, y7, 0, 0, y3, 0]

  p = - s 3 + s 8


 
1215 . Coloring, {2, 4, 7, 10, 11, 12}

R: [7, 8, 7, 7, A, A, A, C, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y7, y4, y5, 0, 0, y6, 3 y4]

  p = - s 3 + s 8


 
1216 . Coloring, {2, 4, 8, 9, 10, 11}

R: [7, 8, 7, 7, A, A, B, B, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y6, y7, 0, y8, y5, 0, y4, y3, y2]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1217 . Coloring, {2, 4, 8, 9, 10, 12}

R: [7, 8, 7, 7, A, A, B, B, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, 3 y6, 0, 0, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y7 - 3 y6 - 3 y5 - 3 y4 + 13 y3 - 3 y2 + 13 y1, 3 y1]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y6, y5, y4, y3, y1, y2, 0, y5, y1, -y6 + y4 + y3 + y2]

  p' = - s 4 + s 8   p' = - s 4 + s 5 - s 6 + s 7   p = s 4 - s 5 + s 6 - s 7


 
1218 . Coloring, {2, 4, 8, 9, 11, 12}

R: [7, 8, 7, 7, A, A, B, B, C, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, 7 y5, 0, 0, 7 y4, 7 y3, 7 y2, 14 y3, -7 y5 - 7 y4 + 11 y3 + 9 y2 + 9 y1, 7 y1]

  p = s 2 - s 4 - s 5 + s 7   p' = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[5 y5, 5 y4, 5 y3, 0, 5 y2, -5 y5 + 11 y4 - 5 y3 - 5 y2 + 11 y1 - 5 y9 + 11 y8 - 5 y7 - 5 y6, 5 y1, 5 y9, 0, 5 y8, 5 y7, 5 y6]

  p = s 4 + s 5 + s 6 - s 8 - s 9 - s 10


 
1219 . Coloring, {2, 4, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, A, B, B, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, 0, y4, y5, y6, y6, y3, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y3, 0, y1, 0, y2, -2 y3 + 2 y5, y3, y4, 0, y5, 0, y6]

  p' = - s 3 + s 7   p = - s 3 + s 7


 
1220 . Coloring, {2, 4, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, B, C, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 9 y1 + 9 y3 - 7 y2 - 7 y4 - 7 y5 + 9 y6 - 7 y7, 0, 7 y1, 0, 0, 7 y3, 7 y2, 7 y4, 7 y5, 7 y6, 7 y7]

  p = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y4, y3, y5, y6, 0, y7, y8, y9]


 
1221 . Coloring, {2, 5, 6, 7, 8, 9}

R: [7, 8, 7, 6, 3, 3, A, B, C, C, 1, 5]
B: [6, 7, 8, 7, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y8, 0, y7, 0, y6, y4, y5, y4, 0, y3, y2, y1]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, -y4 + 2 y6 + y3, 0, y1, 0, y4 - y6, y2, y4 - y6, y3, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1222 . Coloring, {2, 5, 6, 7, 8, 10}

R: [7, 8, 7, 6, 3, 3, A, B, B, 2, 1, 5]
B: [6, 7, 8, 7, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 3 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, 2 y2, 0, 5 y2 - 2 y3, 2 y2, 5 y2 - 2 y3, 2 y1, -2 y1 + 2 y3, 2 y2, 2 y3]

  p = - s 3 + s 6   p = - s 3 + s 8   p = - s 3 + s 7   p = - s 3 + s 4   p = - s 3 + s 5


 
1223 . Coloring, {2, 5, 6, 7, 8, 11}

R: [7, 8, 7, 6, 3, 3, A, B, B, C, 4, 5]
B: [6, 7, 8, 7, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[3 y1, 8 y1 - 3 y2 + 8 y3 - 11 y4 - 11 y5 + 8 y6 - 3 y7, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 5 y1 + 5 y3 - 8 y4 - 8 y5 + 5 y6]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
1224 . Coloring, {2, 5, 6, 7, 8, 12}

R: [7, 8, 7, 6, 3, 3, A, B, B, C, 1, 9]
B: [6, 7, 8, 7, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y4, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y8, y7, y6, y5, y6, 0, y4, y3, y2]

  p = - s 6 + s 9


 
1225 . Coloring, {2, 5, 6, 7, 9, 10}

R: [7, 8, 7, 6, 3, 3, A, C, C, 2, 1, 5]
B: [6, 7, 8, 7, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[2 y3, y1, y5, 0, y2, y3, y4, y6, 0, y8, 0, y7]

  p = - s 2 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y5, 0, y3, y4, y3, y2, y1, y6, y7]

  p = s 5 - s 8


 
1226 . Coloring, {2, 5, 6, 7, 9, 11}

R: [7, 8, 7, 6, 3, 3, A, C, C, C, 4, 5]
B: [6, 7, 8, 7, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y6, 2 y4, y1, y2, y3, y4, 0, y5, 0, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y4, y3, 0, 0, 0, y2, y1, y7, 2 y7, y6, y5, 0]

  p = - s 2 + s 8


 
1227 . Coloring, {2, 5, 6, 7, 9, 12}

R: [7, 8, 7, 6, 3, 3, A, C, C, C, 1, 9]
B: [6, 7, 8, 7, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y6, 0, y1, 0, 0, y6, y5, y6, y3, y4, 0, y2]

  p = - s 5 + s 7   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 2 y6, y6, y5, y6, 0, y4, y3, 0]

  p' = s 4 - s 7   p = s 4 - s 7


 
1228 . Coloring, {2, 5, 6, 7, 10, 11}

Ωp(Δ)=0:     p = s - s 2 - 3s 4 + 4s 5 + 8s 7 - 16s 8

R: [7, 8, 7, 6, 3, 3, A, C, B, 2, 4, 5]
B: [6, 7, 8, 7, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y10, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y5, 2 y4, y4, y3, y2, y7, y6]

  p = - s 6 + s 8


 
1229 . Coloring, {2, 5, 6, 7, 10, 12}

R: [7, 8, 7, 6, 3, 3, A, C, B, 2, 1, 9]
B: [6, 7, 8, 7, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 4 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y3, y2, y1, 0, 0, y10, y9, y8, y7, y6, y5, y4]

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y2, -y1 + y3, y1, -y1 + y4, y1, 0, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8


 
1230 . Coloring, {2, 5, 6, 7, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, C, B, C, 4, 9]
B: [6, 7, 8, 7, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y5, 0, y4, y2, y3, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - 2 y7 - y5 + y6, y1, 0, 0, y2, y3, y4, y7, 0, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
1231 . Coloring, {2, 5, 6, 8, 9, 10}

R: [7, 8, 7, 6, 3, 3, B, B, C, 2, 1, 5]
B: [6, 7, 8, 7, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, 2 y5, y3, 0, y2, y5, y1, 2 y2 - 3 y5, 0, 0, y6, y5]

  p' = - s 5 + s 8   p = - s 4 + s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y7, y2, y3, y4, y5, y6]

  p = - s 2 + s 8


 
1232 . Coloring, {2, 5, 6, 8, 9, 11}

R: [7, 8, 7, 6, 3, 3, B, B, C, C, 4, 5]
B: [6, 7, 8, 7, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y5, y6, y7, 0, 0, y4, 3 y7]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y2, y1, 0, 0, 0, y6, y5, y4, y3, y9, y8, y7]


 
1233 . Coloring, {2, 5, 6, 8, 9, 12}

R: [7, 8, 7, 6, 3, 3, B, B, C, C, 1, 9]
B: [6, 7, 8, 7, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-5 y2 - 10 y6 - 5 y1 + 11 y5 - 5 y3 + 11 y4, 0, 5 y2, 0, 0, 5 y6, 5 y1, 5 y6, 5 y5, 0, 5 y3, 5 y4]

  p' = s 3 + s 4 - s 6 - s 7   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, 0, y5 + y4, y2, y5, y1, y5, 0, y6, y5, y4]

  p = - s 4 + s 7   p' = s 5 - s 8   p' = - s 4 + s 7


 
1234 . Coloring, {2, 5, 6, 8, 10, 11}

R: [7, 8, 7, 6, 3, 3, B, B, B, 2, 4, 5]
B: [6, 7, 8, 7, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y7, y6, y5, y7, y4, y3, y2, 0, 0, y1, 0]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, 0, 0, 0, y2, 2 y1, y1, y3, y4, 0, y5]

  p' = - s 4 + s 6   p = - s 4 + s 6


 
1235 . Coloring, {2, 5, 6, 8, 10, 12}

R: [7, 8, 7, 6, 3, 3, B, B, B, 2, 1, 9]
B: [6, 7, 8, 7, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 5 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y2, 2 y4, y1, 0, 0, y4, y3, 2 y1 - 3 y4, 2 y4, 0, y5, 0]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y3, y1, y3, y2, y3, 0, y4, 0, y5]

  p' = s 3 - s 6   p = - s 3 + s 6


 
1236 . Coloring, {2, 5, 6, 8, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, B, B, C, 4, 9]
B: [6, 7, 8, 7, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, y3, y2, 0, y1, 2 y3]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y3, y1, 0, 0, y2, -3 y3 + 2 y4, y5, y3, 0, y6, 0, y4]

  p = s 4 - s 7   p' = - s 4 + s 7


 
1237 . Coloring, {2, 5, 6, 9, 10, 11}

R: [7, 8, 7, 6, 3, 3, B, C, C, 2, 4, 5]
B: [6, 7, 8, 7, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y5, y4, y3, y1, y2, y6, y7, 0, 0, y8, y9]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8


 
1238 . Coloring, {2, 5, 6, 9, 10, 12}

R: [7, 8, 7, 6, 3, 3, B, C, C, 2, 1, 9]
B: [6, 7, 8, 7, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[11 y1 - 16 y2 - 7 y3 + 9 y4 - 7 y5 + 9 y6, 14 y2, 7 y1, 0, 0, 7 y2, 7 y3, 14 y1 - 21 y2, 7 y4, 0, 7 y5, 7 y6]

  p = s 3 - s 9   p' = s 4 + s 5 - s 7 - s 8   p' = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y3, y2, y1, y2, 0, y6, y7, y5]

  p = - s 5 + s 8


 
1239 . Coloring, {2, 5, 6, 9, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, C, C, C, 4, 9]
B: [6, 7, 8, 7, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, 0, -7 y1 - 7 y2 - 7 y5 + 9 y4 + 9 y3 - 7 y6 + 9 y7, 7 y1, 0, 7 y2, 7 y5, 7 y4, 7 y3, 0, 7 y6, 7 y7]

  p = - s 2 - s 3 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y5, y3, y4, y5, 0, y6, y7, 0]

  p = - s 5 + s 8


 
1240 . Coloring, {2, 5, 6, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, C, B, 2, 4, 9]
B: [6, 7, 8, 7, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y5, 0, y4, y3, y8, y9, 0, y6, y7]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y4, 0, 0, 0, y2, y3, 2 y1, y1, 0, y5, y6, y7]

  p = s 5 - s 8


 
1241 . Coloring, {2, 5, 7, 8, 9, 10}

R: [7, 8, 7, 6, 3, A, A, B, C, 2, 1, 5]
B: [6, 7, 8, 7, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y1, -y2 + y1 + y6 + y5 - y4 - y3 + y8, 0, y6, y7, y5, y4, 0, y3, y8, y7]

  p = s 4 - s 10   p' = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y3, 0, y5, y2, y1, y6, y5, y8, y7]

  p = - s 6 + s 9


 
1242 . Coloring, {2, 5, 7, 8, 9, 11}

R: [7, 8, 7, 6, 3, A, A, B, C, C, 4, 5]
B: [6, 7, 8, 7, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 10 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y9, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y5, y3, y4, 0, 0, y2, y1, y9, y10, y8, y6, y7]


 
1243 . Coloring, {2, 5, 7, 8, 9, 12}

R: [7, 8, 7, 6, 3, A, A, B, C, C, 1, 9]
B: [6, 7, 8, 7, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, y5, y4, y5, y2, y3, y6, y7]

  p' = s 6 - s 8   p = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, y10]


 
1244 . Coloring, {2, 5, 7, 8, 10, 11}

R: [7, 8, 7, 6, 3, A, A, B, B, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y7, 0, y8, 0, 0, y6, 2 y2, y4, y5, y2, y3, y1]

  p = s 7 - s 9


 
1245 . Coloring, {2, 5, 7, 8, 10, 12}

R: [7, 8, 7, 6, 3, A, A, B, B, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y5, y4, y3, 0, 0, y3, y2, y1, 2 y3, y7, y6, 0]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 5 y5 - y1 - y2 - y3 - y4 - y6, y5, y1, y2, y5, y3, 0, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1246 . Coloring, {2, 5, 7, 8, 11, 12}

R: [7, 8, 7, 6, 3, A, A, B, B, C, 4, 9]
B: [6, 7, 8, 7, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y6, y1, 0, y7, y8, y6, y5, y2, y3, y4]

  p = s 3 - s 9

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y1 - y2 + y6 + y5 - y3 + y4 - y7 + y8 - y9, y1, y2, 0, y6, y5, y3, y4, 0, y7, y8, y9]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
1247 . Coloring, {2, 5, 7, 9, 10, 11}

R: [7, 8, 7, 6, 3, A, A, C, C, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y9, y8, y7, y6, y5, y4, y3, 0, y2, 0, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y8, 2 y5, y7, y6, y5, y4, y3]

  p = s 4 - s 9


 
1248 . Coloring, {2, 5, 7, 9, 10, 12}

R: [7, 8, 7, 6, 3, A, A, C, C, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y6, y1, y6, 0, 0, y6, y5, y2, y3, y4, 0, y7]

  p = - s 6 + s 8   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 3 y2, -3 y2 - 7 y1 - 7 y5 + 14 y4 + 8 y3, -5 y1 - 5 y5 + 7 y4 + 7 y3, 3 y6, 3 y1, 3 y5, 0, 3 y4, -3 y6 - 7 y1 - 7 y5 + 8 y4 + 14 y3, 3 y3]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1249 . Coloring, {2, 5, 7, 9, 11, 12}

R: [7, 8, 7, 6, 3, A, A, C, C, C, 4, 9]
B: [6, 7, 8, 7, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, -3 y5 + 2 y4, y4, y5, y3, y2, 0, y1]

  p' = - s 5 + s 7   p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, 0]


 
1250 . Coloring, {2, 5, 7, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, A, C, B, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y7, y8, y9, 0, y10, y1, y2, y3, y4, y5, y6]

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[3 y5, 0, -3 y5 + 5 y1 - 3 y2 - 3 y3 - 3 y4 + 5 y8 - 3 y6 + 5 y7, 0, 3 y1, 3 y2, 3 y3, 3 y4, 0, 3 y8, 3 y6, 3 y7]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1251 . Coloring, {2, 5, 8, 9, 10, 11}

R: [7, 8, 7, 6, 3, A, B, B, C, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y5, y4, y3, y2, y1, y5 + y4 - y3 - y2 + y1 - y9 - y8 + y7 + y6, y9, 0, y8, y7, y6]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1252 . Coloring, {2, 5, 8, 9, 10, 12}

R: [7, 8, 7, 6, 3, A, B, B, C, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y3 - 3 y4 - 3 y2 + 13 y6 - 3 y7 - 3 y5 + 13 y8, 3 y3, 0, 0, 3 y3, 3 y4, 3 y2, 3 y6, 3 y7, 3 y5, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9   p' = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y2, y2 + y4, y1, y4, y2 + y6, y6, 0, y5, y4, y3]

  p = - s 4 + s 7   p' = - s 5 + s 8   p' = - s 4 + s 7


 
1253 . Coloring, {2, 5, 8, 9, 11, 12}

R: [7, 8, 7, 6, 3, A, B, B, C, C, 4, 9]
B: [6, 7, 8, 7, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y3, y4, 0, y5, -y4 + y5 - y1 - y2 + y6 + y7, y3, y1, y2, y6, y7]

  p = - s 7 + s 8   p = - s 7 + s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y10, 0, y6, y7, y8, y9, 0, y5, y3, y4]


 
1254 . Coloring, {2, 5, 8, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, B, B, B, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y7, y6, y5, 0, y4, y3, y2, 2 y6, y1, -y7 + y6 + y5 - y4 + y3 + y2 + y1, 0]

  p' = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = s 3 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, y6, 0, y5, y4, y7, y3, 0, y2, 0, y1]

  p = s 5 - s 8


 
1255 . Coloring, {2, 5, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, B, C, C, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   10

See Matrix
 

[0, -y6 + y7 - y4 + y5 + y3 + y1 + y2 - y9 - y8, y6, y7, 0, y4, y5, y3, y1, y2, y9, y8]

  p = - s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[7 y1, 0, 7 y2, 0, 9 y1 + 9 y2 + 9 y4 - 7 y5 + 9 y3 - 7 y6 + 9 y7 - 7 y8, 7 y4, 7 y5, 7 y3, 0, 7 y6, 7 y7, 7 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1256 . Coloring, {2, 6, 7, 8, 9, 10}

R: [7, 8, 7, 6, A, 3, A, B, C, 2, 1, 5]
B: [6, 7, 8, 7, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y7 + y3 - y4 - y5 + y6, y1, y2 - y7, 0, y2, y7, y3, y4, 0, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9   p' = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y1, 0, y5, y2, y4, y3, y4, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1257 . Coloring, {2, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 4 + 4s 5 - 8s 6 - 8s 7 - 16s 8

R: [7, 8, 7, 6, A, 3, A, B, C, C, 4, 5]
B: [6, 7, 8, 7, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y2, y1, y7, y3, y4, y5, 0, y6, y9, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5 - y6 - y7 + y8 + y9, y1, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
1258 . Coloring, {2, 6, 7, 8, 9, 12}

R: [7, 8, 7, 6, A, 3, A, B, C, C, 1, 9]
B: [6, 7, 8, 7, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y6, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 6 + s 8   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 11 y1 - 5 y2 + 11 y3 - 5 y6 - 5 y4 + 11 y5 - 5 y7 - 5 y8 + 11 y9, 5 y1, 5 y2, 5 y3, 5 y6, 5 y4, 5 y5, 0, 5 y7, 5 y8, 5 y9]

  p = s 4 + s 5 + s 6 - s 8 - s 9 - s 10


 
1259 . Coloring, {2, 6, 7, 8, 10, 11}

R: [7, 8, 7, 6, A, 3, A, B, B, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-y1 + y2 + y3 - y4 - y5 + y6, 0, y1, 0, 0, y2, 2 y1, y3, y4, y5, 2 y3, y6]

  p = - s 6 + s 8   p = - s 6 + s 9   p = - s 6 + s 7


 
1260 . Coloring, {2, 6, 7, 8, 10, 12}

R: [7, 8, 7, 6, A, 3, A, B, B, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y4, y7, y8, 0]

  p = s 3 - s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, y6, y5, y4, y3, y5, y2, 0, y1, y5, -y6 - y4 - y3 - y2 - y1 + 5 y5]

  p' = s 4 - s 8   p' = - s 3 + s 7   p = s 3 - s 7


 
1261 . Coloring, {2, 6, 7, 8, 11, 12}

R: [7, 8, 7, 6, A, 3, A, B, B, C, 4, 9]
B: [6, 7, 8, 7, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y9, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[3 y3, 5 y1 - 3 y2 + 5 y4 - 3 y6, 3 y1 + 3 y4 - 3 y7, 0, 3 y1, 3 y2, -3 y3 + 5 y1 + 5 y4 - 3 y5, 3 y4, 0, 3 y5, 3 y6, 3 y7]

  p = s + s 3 - s 7 - s 9   p' = s - s 2 + 2s 3 - 2s 4 + 2s 5 - 2s 6 + s 7 - s 8   p' = - s 2 + s 3 - 2s 4 + 2s 5 - 2s 6 + 2s 7 - s 8 + s 9


 
1262 . Coloring, {2, 6, 7, 9, 10, 11}

R: [7, 8, 7, 6, A, 3, A, C, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y9, y7, y8, y6, y4, y5, y3, 0, y2, 0, y1]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y2, 2 y1, y3, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


 
1263 . Coloring, {2, 6, 7, 9, 10, 12}

R: [7, 8, 7, 6, A, 3, A, C, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[2 y8, y1, y2, 0, 0, y8, y3, y4, y5, y6, 0, y7]

  p = - s 7 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
1264 . Coloring, {2, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s + s 2 + s 4 + 4s 5 - 8s 7 - 16s 8

R: [7, 8, 7, 6, A, 3, A, C, C, C, 4, 9]
B: [6, 7, 8, 7, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y1 - y2 + y6 - y5 + y3 + y4 + y7 - y8, 0, y6, y5, y3, y4, 0, y7, y8, 0]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1265 . Coloring, {2, 6, 7, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, A, C, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[0, y2, y1, y4, 0, y3, y9, y10, y8, y6, y7, y5]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y4 - y5 - y6 + y7 + y8, 0, y1, 0, y2, y3, y4, y5, 0, y6, y7, y8]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1266 . Coloring, {2, 6, 8, 9, 10, 11}

R: [7, 8, 7, 6, A, 3, B, B, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y5, 0, y6, 0, 0, y5 - 3 y6 + y1 + y2 + y3 - y4 - y7, 2 y6, y1, y2, y3, y4, y7]

  p = s 3 - s 9   p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1267 . Coloring, {2, 6, 8, 9, 10, 12}

R: [7, 8, 7, 6, A, 3, B, B, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 5 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 0, 3 y6 - 3 y5, 3 y4, 3 y3, 3 y2, 3 y6 - 3 y5, -3 y7 - 9 y6 + 3 y5 - 3 y4 - 3 y3 + 13 y2 + 13 y1, 3 y1]

  p = - s 4 + s 10   p = - s 4 - s 5 + s 7 + s 8   p' = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y5 - y2, -y1 + y4 + y5, y1, y4, y2, -y3 + y4 + y5, 0, y3, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8   p = - s 5 + s 9


 
1268 . Coloring, {2, 6, 8, 9, 11, 12}

R: [7, 8, 7, 6, A, 3, B, B, C, C, 4, 9]
B: [6, 7, 8, 7, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -3 y1 - 3 y2 - 3 y3 - 3 y5 + 10 y6 - 3 y4, 3 y1, 0, 3 y2, 3 y3, 3 y5, -3 y5 + 3 y6, 3 y5, 3 y4, 3 y6]

  p = s 2 - s 7   p' = s 2 - s 7   p' = s 3 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[5 y3, -5 y3 + 11 y2 + 11 y1 - 5 y9 - 5 y8 + 11 y7 - 5 y6 - 5 y5 + 11 y4, 5 y2, 0, 5 y1, 5 y9, 5 y8, 5 y7, 0, 5 y6, 5 y5, 5 y4]

  p = s 4 + s 5 + s 6 - s 8 - s 9 - s 10


 
1269 . Coloring, {2, 6, 8, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, B, B, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 4 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y7, y7, y8, 0]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[-y2 + y4, 0, -y3 + y2, 0, y2, y3, -y2 + y4, y1, 0, -y1 + y4, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8


 
1270 . Coloring, {2, 6, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, B, C, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 6 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, y4 + y5 + y7 - y8, y1, -y1 - y3 - y2 + 2 y4 + 2 y5 + 2 y7 - y6, 0, y3, y2, y4, y5, y7, y6, y8]

  p = - s 4 + s 9   p' = - s 4 + s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[-y1 + y5 + y6, 0, y5 + y6 - y3 - y2, 0, y1, y3, y2, -y4 + y5 + y6, 0, y4, y5, y6]

  p = s 2 - s 5 + s 6 - s 9   p' = s 3 - s 4 + s 7 - s 8   p' = s 2 - s 4 + s 6 - s 8


 
1271 . Coloring, {2, 7, 8, 9, 10, 11}

R: [7, 8, 7, 6, A, A, A, B, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y1 + y2 + y3 - y4 - y7 + y5 + y6, y2, y3, 2 y6, y4, 0, y7, y5, y6]

  p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y5, 0, y4, 0, 0, y3, y1, y2, y8, 0, y7, y6]


 
1272 . Coloring, {2, 7, 8, 9, 10, 12}

R: [7, 8, 7, 6, A, A, A, B, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 + 8 y3 - 3 y5 + 5 y7, 3 y4, 0, 0, 0, -3 y4 - 3 y2 + 5 y3 - 3 y6 + 8 y7, 3 y2, 3 y1, 3 y3, 3 y5, 3 y6, 3 y7]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 7 y2, 7 y1, -7 y2 + 9 y1 - 7 y7 + 9 y6 - 7 y5 + 9 y4 - 7 y3, 7 y7, 7 y6, 7 y5, 0, 0, 7 y4, 7 y3]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8


 
1273 . Coloring, {2, 7, 8, 9, 11, 12}

R: [7, 8, 7, 6, A, A, A, B, C, C, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 2 y3, y3, y4, y5, y6, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, y3, y2, 0, y7, y4, y5, y6, 0, 0, y9, y8]


 
1274 . Coloring, {2, 7, 8, 10, 11, 12}

R: [7, 8, 7, 6, A, A, A, B, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y6, 0, y7, 0, y5, y2, y1, y2, y3, y4, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y4, y5, y6, y7, 0, 0, y3, y8]


 
1275 . Coloring, {2, 7, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, A, A, C, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, 0, y3, 0, y2, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y3, 0, y2, 0, y1, y7, y6, y5, 0, 0, y4, y6]

  p = s 3 - s 8


 
1276 . Coloring, {2, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, A, B, B, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, 3 y3, 0, 3 y4, 0, -3 y3 + 5 y2 - 3 y6 + 8 y7, -3 y4 - 3 y1 + 8 y2 - 3 y5 + 5 y7, 3 y1, 3 y2, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y3, 0, y2, 0, y1, y4, 2 y7, y5, 0, 2 y3 - 2 y7, y7, y6]

  p' = s 4 - s 8   p = s 4 - s 8


 
1277 . Coloring, {3, 4, 5, 6, 7, 8}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, C, 1, 5]
B: [6, 8, 7, 6, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1 - y8 + y6 + y7 + y4 + y5 - y2 - y3, 0, y1, 0, y8, y6, y7, y4, y5, y2, y3]

  p = s 8 - s 9


 
1278 . Coloring, {3, 4, 5, 6, 7, 9}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, C, 1, 5]
B: [6, 8, 7, 6, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y1, 0, y5, 0, y4, 0, y3, y2, 0, y6, 0, y7]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3 - y2 + 3 y1 + y4 + y5 - y6, 0, y3, 0, y2, y1, y4, 2 y1, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8


 
1279 . Coloring, {3, 4, 5, 6, 7, 10}

R: [7, 7, 8, 7, 3, 3, A, C, B, 2, 1, 5]
B: [6, 8, 7, 6, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[3 y5, 3 y4, 3 y3, 0, 3 y2, 0, 3 y1, 3 y3 - 3 y2 + 3 y6, 0, -3 y5 - 3 y4 + 10 y3 - 3 y1 - 3 y7 + 10 y6, 3 y7, 3 y6]

  p' = - s 3 - s 5 + s 6 + s 8   p = s 3 + s 5 - s 6 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, 0, y5, y4, y4, y3, y2, y1, y6 - y5 + 2 y4 + y3 + y2 - y1]

  p = - s 6 + s 8   p = - s 6 + s 7


 
1280 . Coloring, {3, 4, 5, 6, 7, 11}

R: [7, 7, 8, 7, 3, 3, A, C, B, C, 4, 5]
B: [6, 8, 7, 6, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y3, y4, y2, 0, y1, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[3 y7, 3 y6, 0, 0, 0, -3 y6 + 8 y1 - 3 y5 + 5 y4, -3 y7 - 3 y3 + 5 y1 - 3 y2 + 8 y4, 3 y3, 3 y1, 3 y2, 3 y5, 3 y4]

  p' = - s 2 + s 8   p = s 2 - s 8


 
1281 . Coloring, {3, 4, 5, 6, 7, 12}

R: [7, 7, 8, 7, 3, 3, A, C, B, C, 1, 9]
B: [6, 8, 7, 6, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, 0, y5, y3, y4, y8, y6, y7]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y4, y8, y5, 0, y6, y7, y8]

  p = - s 3 + s 9


 
1282 . Coloring, {3, 4, 5, 6, 8, 9}

R: [7, 7, 8, 7, 3, 3, B, B, C, C, 1, 5]
B: [6, 8, 7, 6, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y2, 0, y7, y3, y4, y5, y6, y8, y9]


 
1283 . Coloring, {3, 4, 5, 6, 8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, 2, 1, 5]
B: [6, 8, 7, 6, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y3, y1, 0, y3, 0, y4, y5, 0, 0, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y4, 0, y1, y4, y4, y2, y3, 0, y5]

  p' = - s 4 + s 6   p = s 4 - s 6


 
1284 . Coloring, {3, 4, 5, 6, 8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, C, 4, 5]
B: [6, 8, 7, 6, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, 0, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y3, 0, 0, 0, y1, y2, y7, y5, y6, 0, y4]

  p = - s 6 + s 8


 
1285 . Coloring, {3, 4, 5, 6, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, C, 1, 9]
B: [6, 8, 7, 6, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y5, 0, -2 y3 + 2 y2, 0, 0, 0, y4, y3, y2, 0, y1, -2 y3 + 2 y2]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, 2 y5, y3, y4, y5, y2, 0, y6, 0, y7]

  p = s 3 - s 8


 
1286 . Coloring, {3, 4, 5, 6, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, 2, 1, 5]
B: [6, 8, 7, 6, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[-7 y2 + 9 y1 + 9 y5 - 7 y3 + 9 y4 - 7 y6 + 9 y7, 7 y2, 7 y1, 0, 7 y5, 0, 7 y3, 7 y4, 0, 0, 7 y6, 7 y7]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, y3, y4, y5, y6, y7]

  p = s 2 - s 8


 
1287 . Coloring, {3, 4, 5, 6, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, C, 4, 5]
B: [6, 8, 7, 6, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 0, 7 y2, 7 y1, -7 y2 + 9 y1 + 9 y3 - 7 y5 + 9 y4 - 7 y6, 0, 7 y3, 7 y5, 0, 0, 7 y4, 7 y6]

  p = - s - s 2 - s 3 + s 5 + s 6 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y7, y5, y6, 2 y5, y4, y3, 0]

  p = - s 2 + s 8


 
1288 . Coloring, {3, 4, 5, 6, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, C, 1, 9]
B: [6, 8, 7, 6, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[7 y1, 0, 9 y1 + 9 y2 - 7 y3 - 7 y6 + 9 y4 - 7 y5, 0, 0, 0, 7 y2, 7 y3, 7 y6, 0, 7 y4, 7 y5]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y5, 0, y4, 2 y2, y1, y2, y3, 0, y6, y7, 0]

  p = - s 2 + s 8


 
1289 . Coloring, {3, 4, 5, 6, 10, 11}

R: [7, 7, 8, 7, 3, 3, B, C, B, 2, 4, 5]
B: [6, 8, 7, 6, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 3 y6, 3 y5, 3 y4, 3 y3, 0, 3 y2, 3 y1, 0, 0, -3 y6 - 3 y4 - 3 y2 + 10 y3 + 10 y1, -3 y5 + 3 y3 + 3 y1]

  p = s 2 + s 4 - s 5 - s 7   p' = s 2 + s 4 - s 5 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y4, y4, y3, y5, y6, y7]

  p = - s 6 + s 8


 
1290 . Coloring, {3, 4, 5, 6, 10, 12}

R: [7, 7, 8, 7, 3, 3, B, C, B, 2, 1, 9]
B: [6, 8, 7, 6, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y2, 0, 0, 0, y6, y7, y5, 0, y3, y4]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y6, y6, 0, y4, y5, y7]

  p = s 5 - s 8


 
1291 . Coloring, {3, 4, 5, 6, 11, 12}

R: [7, 7, 8, 7, 3, 3, B, C, B, C, 4, 9]
B: [6, 8, 7, 6, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y6, 0, 0, y7, y4, y3, 0, y2, y1]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y1 - y2 - y3 + y4 + y5 - y6, 0, 0, y2, y3, y7, y4, 0, y5, y6, y7]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1292 . Coloring, {3, 4, 5, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, C, 1, 5]
B: [6, 8, 7, 6, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y4 + y5 - y1, 0, -y2 + y4 + y5, 0, y1, 0, y2, -y3 + y4 + y5, 0, y3, y4, y5]

  p = s - s 2 + s 5 - s 6   p' = - s + s 2 - s 5 + s 6   p' = - s + s 3 - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, y7, y8, y9, y10]


 
1293 . Coloring, {3, 4, 5, 7, 8, 10}

R: [7, 7, 8, 7, 3, A, A, B, B, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y4, y5, 0, y6, 0, y1, y2, 0, y7, y8, 0]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -y1 - y2 - y3 - y4 + 2 y5 + 4 y6, y1, 0, y2, y3, y6, y5, y6, y4, y5 + 2 y6]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8


 
1294 . Coloring, {3, 4, 5, 7, 8, 11}

R: [7, 7, 8, 7, 3, A, A, B, B, C, 4, 5]
B: [6, 8, 7, 6, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y5, y4, y3, 0, -y5 + y4 + y3, y2, 0, y4 + y3 - y2, y1, y4 + y3 - y1]

  p = - s + s 4 - s 5 + s 8   p = - s + s 3 - s 5 + s 7   p = - s + s 2 - s 5 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-y1 - y2 - y3 + 2 y4 + 2 y5 + 2 y6 - y7, y4 + y5 + y6 - y8, y1, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p' = - s 4 + s 9   p = s 4 - s 9


 
1295 . Coloring, {3, 4, 5, 7, 8, 12}

R: [7, 7, 8, 7, 3, A, A, B, B, C, 1, 9]
B: [6, 8, 7, 6, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y1 - y2 + y3 + y4 + y5 - y6 - y7, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1 + y2 - y3, y2 + y1 - y5, -y2 + y3 + y4, y1 - y5 + y4, y1, y2, y3 - y5 + y4, 0, y3, y4, y5]

  p' = s - s 6   p' = s 2 - s 7   p' = s 4 - s 9   p = s - s 6   p' = s 3 - s 8


 
1296 . Coloring, {3, 4, 5, 7, 9, 10}

R: [7, 7, 8, 7, 3, A, A, C, C, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[-3 y1 + 10 y2 - 3 y3 + 10 y4 - 3 y5, 3 y1, 3 y2 + 3 y4 - 3 y6, 0, 3 y2, 0, 3 y3, 3 y4, 0, 3 y5, 0, 3 y6 ]

  p = - s 2 - s 4 + s 5 + s 7   p' = - s 2 - s 4 + s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y2, y1, 0, y3, y4, y6, y5, y6, y7, y8]

  p = - s 4 + s 9


 
1297 . Coloring, {3, 4, 5, 7, 9, 11}

R: [7, 7, 8, 7, 3, A, A, C, C, C, 4, 5]
B: [6, 8, 7, 6, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y4 - y2, -y1 + y4, y1, 0, y2, y4 - y3, 0, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y8, 2 y6, y6, y7, 0]

  p = - s 4 + s 9


 
1298 . Coloring, {3, 4, 5, 7, 9, 12}

R: [7, 7, 8, 7, 3, A, A, C, C, C, 1, 9]
B: [6, 8, 7, 6, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1 - 4 y2 - 4 y3 + 4 y4, 0, y1 - 2 y2 - 2 y3 + 2 y4, 0, 0, 0, y1, y2 + y3 - y4, y2, y3, 0, y4]

  p = - s 4 + s 6   p = - s 4 + s 5   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, 0]


 
1299 . Coloring, {3, 4, 5, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 5 y2, 5 y1, -5 y2 + 11 y1 + 11 y5 - 5 y4 + 11 y3 - 5 y7 - 5 y6 + 11 y8, 5 y5, 0, 5 y4, 5 y3, 0, 5 y7, 5 y6, 5 y8]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-3 y1 - 3 y3 - 3 y2 + 3 y5 + 7 y6 - 3 y4, 0, 3 y1, 0, 0, 3 y3, 3 y2, -3 y5 + 3 y6, 3 y5, -3 y5 + 3 y6, 3 y4, 3 y6]

  p' = - s 2 + s 7   p' = - s 3 + s 8   p = - s 2 + s 7


 
1300 . Coloring, {3, 4, 5, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, 0, 0, y7, y8, y9, y4, y5, y6]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 0, 3 y2, -3 y3 - 3 y4 + 5 y5 - 3 y6 - 3 y7 - 3 y8 + 5 y2 + 5 y1, 3 y1]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1301 . Coloring, {3, 4, 5, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, C, 4, 9]
B: [6, 8, 7, 6, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, 0, y5, y4, y3, y2, y1, -y7 + y6 - y5 + y4 + y3 + y2 - y1]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y4, y10, y9, y8, 0, y6, y7, y5]


 
1302 . Coloring, {3, 4, 5, 8, 9, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y3, y2, 0, y3, 0, y4, y6, 0, y7, y5, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y8, y7, 0, y5, y4, y6, y3, y2, y9, y1]


 
1303 . Coloring, {3, 4, 5, 8, 9, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, C, 4, 5]
B: [6, 8, 7, 6, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y6, 0, y5, y4, 0, y3, y8, y7]

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[y3, y2, y1, 0, 0, y10, y9, y8, y7, y6, y5, y4]


 
1304 . Coloring, {3, 4, 5, 8, 9, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, C, 1, 9]
B: [6, 8, 7, 6, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y2, 0, 3 y5, 0, 0, 0, 3 y1, -3 y2 - 3 y5 - 3 y1 + 10 y3 - 3 y4, -3 y5 + 3 y3, 3 y5, 3 y4, 3 y3]

  p' = - s 4 + s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y8, 0, y7, y9, y10]


 
1305 . Coloring, {3, 4, 5, 8, 10, 11}

R: [7, 7, 8, 7, 3, A, B, B, B, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, 2 y4 - 3 y5, y3, 2 y5, 0, y1, y2, 0, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y1, 0, y4, 0, 0, y2, y3, y1, y7, y6, 0, y5]

  p = - s 6 + s 8


 
1306 . Coloring, {3, 4, 5, 8, 10, 12}

R: [7, 7, 8, 7, 3, A, B, B, B, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y4 + y3, y4, 0, 0, 0, y2, y3, 2 y4, y4, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, y6, y7, y5, y4, 0, y3, 0, y2]

  p = - s 5 + s 8


 
1307 . Coloring, {3, 4, 5, 8, 11, 12}

R: [7, 7, 8, 7, 3, A, B, B, B, C, 4, 9]
B: [6, 8, 7, 6, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -y5 + y2, y3, 0, 0, y4, y5, y6, -y5 + y2, y1, y2]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y4, y2, 0, y3, y8, y9, y7, 0, y6, 0, y5]


 
1308 . Coloring, {3, 4, 5, 9, 10, 11}

R: [7, 7, 8, 7, 3, A, B, C, C, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -3 y4 - 3 y6 + 10 y3 + 10 y7 - 3 y1 - 3 y2, 3 y3, 3 y4, 3 y5, 0, 3 y6, 3 y3 - 3 y5 + 3 y7, 0, 3 y1, 3 y2, 3 y7]

  p = - s 3 - s 5 + s 6 + s 8   p' = s 3 + s 5 - s 6 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y8, y3, y4, y5, y6, y7, y9]


 
1309 . Coloring, {3, 4, 5, 9, 10, 12}

R: [7, 7, 8, 7, 3, A, B, C, C, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y1, 6 y1 - 7 y2 - 13 y5 + 6 y3 + 6 y4, 3 y1 - 3 y2 - 7 y5 + 3 y3 + 3 y4, 0, 0, 0, y3, 3 y1 - 4 y2 - 6 y5 + 3 y3 + 3 y4, y2, 3 y1 - 3 y2 - 7 y5 + 3 y3 + 3 y4, y4, y5]

  p = - s 3 + s 9   p' = - s 4 + s 7   p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
1310 . Coloring, {3, 4, 5, 9, 11, 12}

R: [7, 7, 8, 7, 3, A, B, C, C, C, 4, 9]
B: [6, 8, 7, 6, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 7 y3, 7 y6, 0, 0, 7 y5, 7 y4, 7 y2, 7 y3, 7 y1, -14 y3 + 9 y6 + 9 y5 - 7 y4 - 7 y2 + 9 y1]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[y1, y2, y3, 0, y7, y6, y4, y5, 0, y9, y8, 0]


 
1311 . Coloring, {3, 4, 5, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, C, B, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5 + y3, y5, y1, 0, 0, y2, y3, y4, y5, y7, y6]

  p = s 5 - s 8   p' = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y4, 0, y3, 0, y2, y1, y5, y6, 0, y8, y9, y7]


 
1312 . Coloring, {3, 4, 6, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, C, 1, 5]
B: [6, 8, 7, 6, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y4, 0, y3, y5, 0, y7, y8, y6]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y5, y4, y6, y7, y8, y9, y10]


 
1313 . Coloring, {3, 4, 6, 7, 8, 10}

R: [7, 7, 8, 7, A, 3, A, B, B, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y3, 0, y4, y5, 0, y6, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y4, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 7 + s 9


 
1314 . Coloring, {3, 4, 6, 7, 8, 11}

R: [7, 7, 8, 7, A, 3, A, B, B, C, 4, 5]
B: [6, 8, 7, 6, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, y4, 0, y3, y2, 0, y1, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   10

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, y9, y10]


 
1315 . Coloring, {3, 4, 6, 7, 8, 12}

R: [7, 7, 8, 7, A, 3, A, B, B, C, 1, 9]
B: [6, 8, 7, 6, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, y10]


 
1316 . Coloring, {3, 4, 6, 7, 9, 10}

R: [7, 7, 8, 7, A, 3, A, C, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y1, y2, y1, 0, y3, 0, y4, y5, 0, y7, 0, y6]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
1317 . Coloring, {3, 4, 6, 7, 9, 11}

R: [7, 7, 8, 7, A, 3, A, C, C, C, 4, 5]
B: [6, 8, 7, 6, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y3, 2 y3, y1, 0, y3 + 2 y2, y2, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y5, y6, y4, 2 y3, y7, y8, 0]

  p = - s 3 + s 9


 
1318 . Coloring, {3, 4, 6, 7, 9, 12}

R: [7, 7, 8, 7, A, 3, A, C, C, C, 1, 9]
B: [6, 8, 7, 6, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1 - 4 y2 - 4 y3 + 4 y4, 0, y1 - 2 y2 - 2 y3 + 2 y4, 0, 0, 0, y1, y2 + y3 - y4, y2, y3, 0, y4]

  p = - s 4 + s 7   p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y6, y3, y4, y5, y9, 0, y7, y8, 0]


 
1319 . Coloring, {3, 4, 6, 7, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y6, y6 + y3, y5, 0, y4, y3, 0, y2, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y7, 0, y6, 0, 0, y5, y4, y6, y3, y1, y2, y8]

  p = s 7 - s 9


 
1320 . Coloring, {3, 4, 6, 7, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y5, y4, y3, 0, 0, 0, y2, y1, y9, y8, y7, y6]

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3, y1, y2, y4, y5, y6, 0, y8, y9, y7]


 
1321 . Coloring, {3, 4, 6, 7, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, C, 4, 9]
B: [6, 8, 7, 6, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y2 + y3 + y4 + y5 - y6 - y7, y1, 0, 0, y2, y3, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y3, y1, 0, y4, y5, y6, y7, 0, y8, y9, y10]


 
1322 . Coloring, {3, 4, 6, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, y3, 0, y5 + y3, 0, y7, y5, 0, y6, y4, y3]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y7, 0, y1, y2, y3, y4, y5, y6, y8]

  p = - s 3 + s 9


 
1323 . Coloring, {3, 4, 6, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, C, 4, 5]
B: [6, 8, 7, 6, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y1, y2, y3, 0, y5, 4 y1 + 4 y2 + 5 y3 - y5 - 6 y4, 0, -2 y1 - 2 y2 - 4 y3 + 5 y4, y4, 5 y1 + 5 y2 + 6 y3 - 8 y4]

  p = - s 3 + s 6   p' = s 4 - s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, y9, y10]


 
1324 . Coloring, {3, 4, 6, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, C, 1, 9]
B: [6, 8, 7, 6, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y3 - 3 y1 - 3 y2 + 10 y5 - 3 y4, 0, 3 y3, 0, 0, 0, 3 y1, 3 y2, -3 y3 + 3 y5, 3 y3, 3 y4, 3 y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y4, y5, y6, y7, y8, y9, y1, 0, y10, y2, y3]


 
1325 . Coloring, {3, 4, 6, 8, 10, 11}

R: [7, 7, 8, 7, A, 3, B, B, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 5 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 2 y5 - y4, y1, 4 y5 - 2 y4, 0, y6, y5, 0, y4, y3, 0 ]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, y2, 0, 0, 2 y1, y1, y2, y5, y4, 0, y3]

  p' = - s 4 + s 6   p = - s 4 + s 6   p = - s 4 + s 8


 
1326 . Coloring, {3, 4, 6, 8, 10, 12}

R: [7, 7, 8, 7, A, 3, B, B, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y4 + y3, y4, 0, 0, 0, y2, y3, 2 y4, y4, y5, 0]

  p' = s 4 - s 7   p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, 2 y5, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 3 + s 8


 
1327 . Coloring, {3, 4, 6, 8, 11, 12}

R: [7, 7, 8, 7, A, 3, B, B, B, C, 4, 9]
B: [6, 8, 7, 6, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, 0, y6, y5, y3, y4, y2, y4 + y5]

  p = - s 4 + s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y3, y2, y1, 0, y4, y6, y5, y7, 0, y8, 0, y9]


 
1328 . Coloring, {3, 4, 6, 9, 10, 11}

R: [7, 7, 8, 7, A, 3, B, C, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y3, y2, y1, y5, 0, y4, y9, 0, y8, y7, y6]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, y5, y6, y4, y2, y3, y8, y7]

  p = s 3 - s 9


 
1329 . Coloring, {3, 4, 6, 9, 10, 12}

R: [7, 7, 8, 7, A, 3, B, C, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y1, 6 y1 - 7 y3 - 13 y5 + 6 y2 + 6 y4, 3 y1 - 3 y3 - 7 y5 + 3 y2 + 3 y4, 0, 0, 0, y2, 3 y1 - 4 y3 - 6 y5 + 3 y2 + 3 y4, y3, 3 y1 - 3 y3 - 7 y5 + 3 y2 + 3 y4, y4, y5]

  p = - s 3 + s 9   p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y8, y4, y5, 0, y6, y7, y9]


 
1330 . Coloring, {3, 4, 6, 9, 11, 12}

R: [7, 7, 8, 7, A, 3, B, C, C, C, 4, 9]
B: [6, 8, 7, 6, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 7 y6, 7 y5, 0, 0, 7 y4, 7 y3, 7 y2, 7 y6, 7 y1, -14 y6 + 9 y5 + 9 y4 - 7 y3 - 7 y2 + 9 y1]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y9, y8, y7, 0, y6, y5, y4, y3, 0, y2, y1, 0]


 
1331 . Coloring, {3, 4, 6, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, C, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4 + y2, y4, y1, 0, 0, y7, y2, y3, y4, y5, y6]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y3, 0, y4, 0, y1, y2, y8, y9, 0, y5, y6, y7]


 
1332 . Coloring, {3, 4, 7, 8, 9, 10}

R: [7, 7, 8, 7, A, A, A, B, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y5 + y6, 0, y3, y6, 0, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y2, y1, 0, y3, y6, y5, y4, 0, y7, y8]


 
1333 . Coloring, {3, 4, 7, 8, 9, 11}

R: [7, 7, 8, 7, A, A, A, B, C, C, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, 0, y3, y4, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y8, 0, y9, y7]


 
1334 . Coloring, {3, 4, 7, 8, 9, 12}

R: [7, 7, 8, 7, A, A, A, B, C, C, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y4, y3, y9, y6, y7, 0, 0, y8, y5]


 
1335 . Coloring, {3, 4, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, 0, y3, y4, 0, y5, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}, {1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[-7 y1 - 7 y3 - 7 y4 + 9 y2 + 9 y5 - 7 y7 + 9 y6, 0, 7 y1, 0, 0, 7 y3, 7 y4, 7 y2, 7 y5, 0, 7 y7, 7 y6]

  p = - s 2 - s 3 + s 7 + s 8


 
1336 . Coloring, {3, 4, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 0, 0, y6, y5, 2 y5, y4, y3, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, 0, y7, y8]


 
1337 . Coloring, {3, 4, 7, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, 0, y8, y9]


 
1338 . Coloring, {3, 4, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y4, 0, 2 y1, y3, 0, y2, y1, 0, y6, 0, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, 0, y4, y5, y6, y7, 0, y3, 2 y6]

  p = - s 3 + s 8


 
1339 . Coloring, {3, 4, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[10 y3 + 10 y2 - 22 y1 + 10 y5 - 22 y4, y3, 0, 0, 0, 0, y2, 5 y3 + 5 y2 - 11 y1 + 5 y5 - 11 y4, y1, y5, 0, y4]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y5, y3, y4, y2, y6, 0, 0, y7, 2 y6]

  p = - s 3 + s 8


 
1340 . Coloring, {3, 4, 7, 9, 11, 12}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, 2 y2, 0, 0, y1, y2, y3, y4, 0, y5]

  p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, y6, y5, 0, y6, y4, y3, y2, 0, 0, y1, 0]

  p = - s 3 + s 8


 
1341 . Coloring, {3, 4, 7, 10, 11, 12}

R: [7, 7, 8, 7, A, A, A, C, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y7, 0, 0, y5, y6, y2, y3, y4, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y6, 0, y4, y5, y2, y3, 0, 0, y7, 3 y3]

  p = - s 3 + s 8


 
1342 . Coloring, {3, 4, 8, 9, 10, 11}

R: [7, 7, 8, 7, A, A, B, B, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y4, y3, 0, y2, y5, 0, y6, y7, y5]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1343 . Coloring, {3, 4, 8, 9, 10, 12}

R: [7, 7, 8, 7, A, A, B, B, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 3 y3 - 9 y2 + 13 y6 - 3 y5 + 13 y4, 0, 0, 0, 0, 3 y3, 3 y2, 3 y6, 6 y2, 3 y5, 3 y4]

  p = - s 3 - s 4 + s 6 + s 7   p' = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y7, y6, y5, y4, y3, y2, 0, y1, y2, y8]

  p = s 4 - s 9


 
1344 . Coloring, {3, 4, 8, 9, 11, 12}

R: [7, 7, 8, 7, A, A, B, B, C, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 10 vs 10

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, -7 y1 + 11 y2 + 9 y3 - 7 y4 + 9 y5, 0, 0, 7 y1, 7 y2, 7 y3, 14 y2, 7 y4, 7 y5]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]


 
1345 . Coloring, {3, 4, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, A, B, B, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, y4, 2 y4, 2 y4, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y5, 0, y1, 0, y2, y3, y4, y5, 0, y7, 0, y6]

  p = - s 3 + s 8


 
1346 . Coloring, {3, 4, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, B, C, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, y6, 0, y5, 0, 0, y4, 5 y6 + 5 y5 + 5 y4 - 11 y2 + 5 y3 - 11 y1, y2, 10 y6 + 10 y5 + 10 y4 - 22 y2 + 10 y3 - 22 y1, y3, y1]

  p = s 3 + s 4 - s 6 - s 7   p' = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y4, y3, y8, y7, 0, y6, y5, y9]


 
1347 . Coloring, {3, 5, 6, 7, 8, 9}

R: [7, 7, 8, 6, 3, 3, A, B, C, C, 1, 5]
B: [6, 8, 7, 7, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y8, 0, y7, 0, y6, y5, y4, y3, 0, y2, y1, y8 - y7 + y6 + y5 - y4 + y3 + y2 - y1]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y2, 0, y1, 0, y5, y6, y3, y4, y7, y8, y9]


 
1348 . Coloring, {3, 5, 6, 7, 8, 10}

R: [7, 7, 8, 6, 3, 3, A, B, B, 2, 1, 5]
B: [6, 8, 7, 7, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 3 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y4, y4, y5, y6, 0, y7, y8, 0]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, 2 y2, 0, 5 y2 - 2 y3, 2 y2, 5 y2 - 2 y3, -2 y1 + 2 y3, 2 y1, 2 y2, 2 y3]

  p' = s 6 - s 7   p' = s 5 - s 7   p' = s 4 - s 7   p' = s 3 - s 7   p = s 3 - s 8


 
1349 . Coloring, {3, 5, 6, 7, 8, 11}

R: [7, 7, 8, 6, 3, 3, A, B, B, C, 4, 5]
B: [6, 8, 7, 7, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1350 . Coloring, {3, 5, 6, 7, 8, 12}

R: [7, 7, 8, 6, 3, 3, A, B, B, C, 1, 9]
B: [6, 8, 7, 7, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y1 + y3 - y2 + y4 + y5 + y6 - y8 - y7, 0, 0, y3, y2, y4, y5, y6, y8, y7]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, 5 y5 - y1 - y2 - y3 - y4 - y6, 0, y5, y1, y2, y5, y3, 0, y4, y5, y6]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8


 
1351 . Coloring, {3, 5, 6, 7, 9, 10}

R: [7, 7, 8, 6, 3, 3, A, C, C, 2, 1, 5]
B: [6, 8, 7, 7, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[2 y1, 2 y3 + 2 y4 - y2 - y5, y3 + y1 + y4 - y6, 0, y3, y1, y2, y4, 0, y5, 0, y6]

  p' = s 2 + s 4 - s 5 - s 7   p' = - s 3 - s 5 + s 6 + s 8   p = s 2 + s 4 - s 5 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, y6, y5, y6, y4, y3, y2, y1]

  p = s 5 - s 8


 
1352 . Coloring, {3, 5, 6, 7, 9, 11}

R: [7, 7, 8, 6, 3, 3, A, C, C, C, 4, 5]
B: [6, 8, 7, 7, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y3, -2 y2 + 2 y4, y1, y2, -2 y2 + 2 y4, y5, 0, y4, 0, y6]

  p = - s 3 + s 7   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, y5, y4, y5, y6, y7, 0]

  p = s 2 - s 8


 
1353 . Coloring, {3, 5, 6, 7, 9, 12}

R: [7, 7, 8, 6, 3, 3, A, C, C, C, 1, 9]
B: [6, 8, 7, 7, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 4 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, y1, 0, 0, y2, 2 y1 - 2 y2, y3, 5 y1 - 9 y2 - 3 y3 + y4, -2 y1 + 4 y2 + 2 y3, 0, y4]

  p = s 4 - s 8   p' = s 6 - s 7   p' = s 5 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, y4, y3, y5, 0, y6, y7, 0]

  p = - s 5 + s 8


 
1354 . Coloring, {3, 5, 6, 7, 10, 11}

R: [7, 7, 8, 6, 3, 3, A, C, B, 2, 4, 5]
B: [6, 8, 7, 7, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {2, 7, 10}}    order:   12

See Matrix
 

[0, 2 y4, 2 y2, 5 y4 - 2 y2 - 2 y5, 5 y4 - 2 y1 - 2 y3 - 2 y6, 2 y1, 2 y4, 2 y3, 0, 2 y4, 2 y6, 2 y5]

  p' = - s 5 + s 9   p = - s 4 - s 6 + s 7 + s 9   p = - s 4 + s 5 - s 6 + s 7   p = - s 4 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, 2 y6, y6, y5, y4, y3, y2]

  p = s 6 - s 8


 
1355 . Coloring, {3, 5, 6, 7, 10, 12}

R: [7, 7, 8, 6, 3, 3, A, C, B, 2, 1, 9]
B: [6, 8, 7, 7, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y10, y9, y8, 0, 0, y7, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y2, -y1 + y3, y1, -y1 + y4, y1, 0, y2, y3, y4]

  p = - s 2 + s 8   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 5


 
1356 . Coloring, {3, 5, 6, 7, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, C, B, C, 4, 9]
B: [6, 8, 7, 7, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y8, 0, y9, y5, y6, y7, y2, y3, y4]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5 + y6 - 3 y7, y1, 0, 0, y2, y3, 2 y7, y4, 0, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
1357 . Coloring, {3, 5, 6, 8, 9, 10}

R: [7, 7, 8, 6, 3, 3, B, B, C, 2, 1, 5]
B: [6, 8, 7, 7, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y6, y2, 0, y3, y6, y5, y4, 0, 0, y7, y6]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y4, y6, y4, y5, y2, y3, y7]

  p = - s 2 + s 8


 
1358 . Coloring, {3, 5, 6, 8, 9, 11}

R: [7, 7, 8, 6, 3, 3, B, B, C, C, 4, 5]
B: [6, 8, 7, 7, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, 2 y3, 2 y2, 2 y1, 2 y4, 2 y5, 2 y6, 0, 0, 2 y7, 3 y5]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y4, y3, y9, y5, y6, y7, y8]


 
1359 . Coloring, {3, 5, 6, 8, 9, 12}

R: [7, 7, 8, 6, 3, 3, B, B, C, C, 1, 9]
B: [6, 8, 7, 7, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y6, 0, 5 y5, 0, 0, 5 y4, 5 y2, 5 y3, 5 y1, 0, -5 y6 - 5 y5 - 5 y4 - 5 y2 - 5 y3 + 11 y1 + 11 y7, 5 y7]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y2, y3, y7, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 9


 
1360 . Coloring, {3, 5, 6, 8, 10, 11}

R: [7, 7, 8, 6, 3, 3, B, B, B, 2, 4, 5]
B: [6, 8, 7, 7, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y3, y1, y2, y3, y4, y5, y6, 0, 0, y7, 0]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, 0, 0, 0, y1, 2 y2, y2, y3, y4, 0, y5]

  p' = s 4 - s 6   p = s 4 - s 6


 
1361 . Coloring, {3, 5, 6, 8, 10, 12}

R: [7, 7, 8, 6, 3, 3, B, B, B, 2, 1, 9]
B: [6, 8, 7, 7, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y3, y2, 0, 0, y3, y4, y5, 2 y3, 0, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y3, y2, y3, y1, y3, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1362 . Coloring, {3, 5, 6, 8, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, B, B, C, 4, 9]
B: [6, 8, 7, 7, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y2, y3, 0, y1, y6, y4, y5, 0, y7, y6]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y3, y4, 0, 0, y1, y2, y3, y5, 0, y6, 0, y7]

  p = s 3 - s 8


 
1363 . Coloring, {3, 5, 6, 9, 10, 11}

R: [7, 7, 8, 6, 3, 3, B, C, C, 2, 4, 5]
B: [6, 8, 7, 7, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y2, y2 - y1 + y8 + y5 - y6 + y7 + y3 - y4, y1, y8, y5, y6, y7, 0, 0, y3, y4]

  p = - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 2 y3, y3, y6, y5, y4, y7]

  p = - s 2 + s 8


 
1364 . Coloring, {3, 5, 6, 9, 10, 12}

R: [7, 7, 8, 6, 3, 3, B, C, C, 2, 1, 9]
B: [6, 8, 7, 7, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-y2 + 2 y3 + 2 y6 - y4, 2 y1, y1 + y3 + y6 - y5, 0, 0, y1, y2, y3, y6, 0, y4, y5]

  p' = s 4 - s 7   p = - s 4 + s 7   p' = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, y3, y4, y3, 0, y7, y5, y6]

  p = s 5 - s 8


 
1365 . Coloring, {3, 5, 6, 9, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, C, C, C, 4, 9]
B: [6, 8, 7, 7, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, 0, y5, y3, y2, y4, 0, y1, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, 0, 0, y5, y4, y5, y2, 0, y3, y1, 0]

  p = s 2 - s 8


 
1366 . Coloring, {3, 5, 6, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, C, B, 2, 4, 9]
B: [6, 8, 7, 7, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y2, y1, y3, 0, y7, y6, y5, y4, 0, y8, y9]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y5, y3, 2 y4, y4, 0, y2, y6, y7]

  p = - s 5 + s 8


 
1367 . Coloring, {3, 5, 7, 8, 9, 10}

R: [7, 7, 8, 6, 3, A, A, B, C, 2, 1, 5]
B: [6, 8, 7, 7, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y4, y7, y5, y6, 0, y8, y9, y7]

  p = s 7 - s 10

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, -2 y1 + y6, y3, -2 y1 + y6, y4, -2 y1 + y6, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1368 . Coloring, {3, 5, 7, 8, 9, 11}

R: [7, 7, 8, 6, 3, A, A, B, C, C, 4, 5]
B: [6, 8, 7, 7, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4 + y6 - y1 - y2, y3, -y3 + y4 + y6, y1, y2, y5, 0, -y5 + y4 + y6, y4, y6]

  p' = - s 2 + s 4 - s 6 + s 8   p = s 2 - s 3 + s 6 - s 7   p' = - s 2 + s 3 - s 6 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[y1, y2, y3, 0, 0, y10, y7, y8, y9, y4, y5, y6]


 
1369 . Coloring, {3, 5, 7, 8, 9, 12}

R: [7, 7, 8, 6, 3, A, A, B, C, C, 1, 9]
B: [6, 8, 7, 7, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[2 y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y1, y2, y3, y4, y5, y6, y7]

  p' = s 7 - s 8   p = s 7 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {2, 5, 8, 10, 12}}   

See Matrix
 

[0, 7 y1, 9 y1 - 7 y2 + 9 y3 - 7 y4 - 7 y5 + 9 y6 + 9 y8 - 7 y7 + 9 y9, 7 y2, 7 y3, 7 y4, 7 y5, 7 y6, 0, 7 y8, 7 y7, 7 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10


 
1370 . Coloring, {3, 5, 7, 8, 10, 11}

R: [7, 7, 8, 6, 3, A, A, B, B, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, 0]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-y6 - y5 - y3 + 4 y4 + 2 y2 - y1, 0, y6, 0, 0, y5, y3, y4, y2, y4, y1, 2 y4 + y2]

  p' = - s 3 + s 8   p' = - s 2 + s 7   p = - s 2 + s 7


 
1371 . Coloring, {3, 5, 7, 8, 10, 12}

R: [7, 7, 8, 6, 3, A, A, B, B, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y3, y4, y7, 2 y3, y5, y6, 0]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -y1 + y2 - y3 + y4, y1, -2 y3 + 2 y4 - y5 + 2 y2, y3 - 2 y4 + y5, y2, y3 - 2 y4 + y5, 0, y3, y4, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9


 
1372 . Coloring, {3, 5, 7, 8, 11, 12}

R: [7, 7, 8, 6, 3, A, A, B, B, C, 4, 9]
B: [6, 8, 7, 7, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 10

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, -3 y7 + y6 + y1 + y5 + y4 - y2 - y3, 2 y7, y1, y5, y4, y2, y3]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1 - y4 + y3, y1 - y2 + y5, y4 + y5 - y3, 0, y1 + y3 - y2, y1, y4 - y2 + y5, y4, 0, y2, y3, y5]

  p' = - s 2 + s 7   p' = - s 3 + s 8   p' = - s 4 + s 9   p = - s + s 6   p' = - s + s 6


 
1373 . Coloring, {3, 5, 7, 9, 10, 11}

R: [7, 7, 8, 6, 3, A, A, C, C, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -3 y1 + 10 y2 - 3 y3 - 3 y4 + 10 y5 - 3 y6, 3 y2 + 3 y5 - 3 y7, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 0, 3 y6, 0, 3 y7]

  p' = s 3 + s 5 - s 6 - s 8   p = s 3 + s 4 + s 5 - s 7 - s 8 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y3, 0, 0, y2, y4, y6, y5, y6, y7, y8]

  p = - s 4 + s 9


 
1374 . Coloring, {3, 5, 7, 9, 10, 12}

R: [7, 7, 8, 6, 3, A, A, C, C, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 5 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[6 y1 + 6 y4 - 20 y5 + 6 y2, y1, 3 y1 + 3 y4 - 10 y5 + 3 y2, 0, 0, 3 y1 + 3 y4 - 10 y5 + 3 y2, y4, y3, 3 y1 + 3 y4 - 9 y5 + 3 y2 - y3, y2, 0, y5]

  p = - s 3 + s 9   p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -3 y1 - 7 y3 - 7 y2 + 14 y4 + 8 y5, 3 y1, -5 y3 - 5 y2 + 7 y4 + 7 y5, 3 y3, 3 y2, 3 y3, 0, 3 y4, -10 y3 - 7 y2 + 8 y4 + 14 y5, 3 y5]

  p' = - s 4 + s 7   p = - s 3 + s 9   p = - s 3 + s 6   p' = - s 3 + s 6


 
1375 . Coloring, {3, 5, 7, 9, 11, 12}

R: [7, 7, 8, 6, 3, A, A, C, C, C, 4, 9]
B: [6, 8, 7, 7, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, -y1 + 2 y2 + 2 y3 - 2 y4, -2 y1 + 4 y2 + 4 y3 - 4 y4, 0, y1, -2 y1 + 4 y2 + 4 y3 - 4 y4, y2 + y3 - y4, y2, y3, 0, y4]

  p = - s 4 + s 6   p = - s 4 + s 7   p = - s 4 + s 8   p = - s 4 + s 5

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, 0]


 
1376 . Coloring, {3, 5, 7, 10, 11, 12}

R: [7, 7, 8, 6, 3, A, A, C, B, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y4, 0, y5, y3, y6, y7, y8, y10, y9]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[-3 y8 + 5 y1 - 3 y7 - 3 y5 - 3 y6 + 5 y3 - 3 y2 + 5 y4, 0, 3 y8, 0, 3 y1, 3 y7, 3 y5, 3 y6, 0, 3 y3, 3 y2, 3 y4]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1377 . Coloring, {3, 5, 8, 9, 10, 11}

R: [7, 7, 8, 6, 3, A, B, B, C, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y5 + y6 + y7 - y8 - y9, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y6, y7, y8, y9, y3, y4, y5]


 
1378 . Coloring, {3, 5, 8, 9, 10, 12}

R: [7, 7, 8, 6, 3, A, B, B, C, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y7 - 3 y6 + 13 y4 - 6 y5 - 3 y3 + 13 y2, 3 y7, 0, 0, 3 y7, 3 y6, 3 y5, 3 y4, 3 y5, 3 y3, 3 y2]

  p = - s 4 + s 10   p = s 4 - s 6 - s 7 + s 9   p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y4, y1, y6, y4, y5, y4, 0, y3, y4, y2]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1379 . Coloring, {3, 5, 8, 9, 11, 12}

R: [7, 7, 8, 6, 3, A, B, B, C, C, 4, 9]
B: [6, 8, 7, 7, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y2 - y3 - y4 - y5 + y6 + y7, y1, 0, y2, -2 y1 + 2 y2 - 2 y3 - 2 y4 - 2 y5 + 2 y6 + 2 y7, y3, y4, y5, y6, y7]

  p' = s 7 - s 8   p = s 7 - s 9

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[y2, y1, y3, 0, y4, y7, y6, y5, 0, y10, y9, y8]


 
1380 . Coloring, {3, 5, 8, 10, 11, 12}

Ωp(Δ)=0:     p = - s + s 2 + 2s 3 - 3s 4 + 4s 5 + 8s 6 - 8s 7 + 16s 8

R: [7, 7, 8, 6, 3, A, B, B, B, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y5 - y2 + y4 + y3 + y6 - y7, y1, y5, 0, y2, y4, y3, 2 y1, y6, y7, 0]

  p' = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y5, 0, y1, 0, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 5 + s 8


 
1381 . Coloring, {3, 5, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - s + s 2 + 2s 3 + 3s 4 - 8s 5 + 8s 7 - 16s 8

R: [7, 7, 8, 6, 3, A, B, C, C, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 10 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 5 y3 + 5 y1 - 3 y4 - 3 y7, 3 y3 + 3 y1 - 3 y6, 5 y3 + 5 y1 - 3 y5 - 3 y2, 0, 3 y4, 3 y5, 3 y3, 3 y1, 3 y2, 3 y7, 3 y6]

  p' = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p' = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y8, y7, y9]


 
1382 . Coloring, {3, 6, 7, 8, 9, 10}

R: [7, 7, 8, 6, A, 3, A, B, C, 2, 1, 5]
B: [6, 8, 7, 7, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y3 + y7, y7, y8, y6, 0, y4, y5, y7]

  p' = - s 6 + s 9   p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y3, y2, y3, y4, y5, y6, y7]

  p' = s 5 - s 8   p = s 5 - s 8


 
1383 . Coloring, {3, 6, 7, 8, 9, 11}

R: [7, 7, 8, 6, A, 3, A, B, C, C, 4, 5]
B: [6, 8, 7, 7, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 0, 3 y1, 3 y2, 5 y1 + 5 y2 + 5 y3 - 3 y4 + 5 y6 - 3 y5 + 5 y7 - 3 y8, 3 y3, 3 y4, 3 y6, 0, 3 y5, 3 y7, 3 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y2, y1, y2 - y1 - y3 + y4 + y7 + y6 + y5 - y9 - y8, 0, 0, y3, y4, y7, y6, y5, y9, y8]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9 + s 10


 
1384 . Coloring, {3, 6, 7, 8, 9, 12}

R: [7, 7, 8, 6, A, 3, A, B, C, C, 1, 9]
B: [6, 8, 7, 7, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y6, y7, y8, y9, y5, y3, y4]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   9

See Matrix
 

[0, y1, y4, y5, y2, y8, y3, y7, 0, y10, y6, y9]


 
1385 . Coloring, {3, 6, 7, 8, 10, 11}

R: [7, 7, 8, 6, A, 3, A, B, B, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 9 y1 + 9 y7 - 7 y6 + 9 y3 - 7 y4 + 9 y5 - 7 y2 + 9 y8, 7 y1, 7 y7, 7 y6, 7 y3, 7 y4, 7 y5, 0, 7 y2, 7 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y2, 0, y5, 0, 0, y2 + y1 + y6 + y7 - y3 - y4, y1, y5, y6, y7, y3, y4]

  p = - s 7 + s 8   p = - s 7 + s 9


 
1386 . Coloring, {3, 6, 7, 8, 10, 12}

R: [7, 7, 8, 6, A, 3, A, B, B, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y6, y4, y5, 2 y6, y7, y8, 0]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y6, y4, 0, y5, y7, y8]

  p = - s 6 + s 9


 
1387 . Coloring, {3, 6, 7, 8, 11, 12}

R: [7, 7, 8, 6, A, 3, A, B, B, C, 4, 9]
B: [6, 8, 7, 7, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 10

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y3, 0, y2, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y3, y2, y1, 0, -y3 + y2 + y1 + y7 - y8 - y6 - y4 + y5 + y9, y7, y8, y6, 0, y4, y5, y9]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10


 
1388 . Coloring, {3, 6, 7, 9, 10, 11}

R: [7, 7, 8, 6, A, 3, A, C, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y5, y4, y6, y7, 0, y9, 0, y8]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y3 - y1 - y6 - y4 + y5 + y7, 0, y2, 0, 0, y3, y1, y2, y6, y4, y5, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9


 
1389 . Coloring, {3, 6, 7, 9, 10, 12}

R: [7, 7, 8, 6, A, 3, A, C, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-14 y1 + 18 y2 - 14 y3 + 18 y4 + 18 y5 - 14 y6 + 18 y7, 5 y1, 5 y2, 0, 0, -7 y1 + 9 y2 - 7 y3 + 9 y4 + 9 y5 - 7 y6 + 9 y7, 5 y3, 5 y4, 5 y5, 5 y6, 0, 5 y7]

  p = s 4 - s 6 - s 7 + s 9   p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y4, y2, y3, y1, y3, 0, y8, y6, y7]

  p = - s 6 + s 9


 
1390 . Coloring, {3, 6, 7, 9, 11, 12}

R: [7, 7, 8, 6, A, 3, A, C, C, C, 4, 9]
B: [6, 8, 7, 7, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y4, y3, 0, -y3 + y6, y3, y2, y1, y6, 0, y5]

  p' = - s 5 + s 7   p = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 + y6 - y4 - y5 - y7 + y8, y1, y2, 0, y3, y6, y4, y5, 0, y7, y8, 0]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1391 . Coloring, {3, 6, 7, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, A, C, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 9, 11, 12}}   

See Matrix
 

[0, 7 y1, 9 y1 - 7 y2 - 7 y3 + 9 y8 - 7 y4 - 7 y5 + 9 y6 - 7 y7 - 7 y9, 7 y2, 0, 7 y3, 7 y8, 7 y4, 7 y5, 7 y6, 7 y7, 7 y9]

  p = - s - s 2 - s 3 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y4 - y5 - y6 + y7 + y8, 0, y1, 0, y2, y3, y4, y5, 0, y6, y7, y8]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1392 . Coloring, {3, 6, 8, 9, 10, 11}

R: [7, 7, 8, 6, A, 3, B, B, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, y7, 0, y8, y9, y10]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
1393 . Coloring, {3, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 4 - 4s 5 - 8s 7 + 16s 8

R: [7, 7, 8, 6, A, 3, B, B, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 0, 3 y6 - 3 y5, 3 y4, 3 y3, 3 y2, 3 y6 - 3 y5, -3 y7 - 9 y6 + 3 y5 - 3 y4 - 3 y3 + 13 y2 + 13 y1, 3 y1]

  p = s 4 + s 5 - s 7 - s 8   p' = s 4 - s 6 - s 7 + s 9   p' = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y3, y4, y6, y2, y6, 0, y7, y6, y5]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
1394 . Coloring, {3, 6, 8, 9, 11, 12}

R: [7, 7, 8, 6, A, 3, B, B, C, C, 4, 9]
B: [6, 8, 7, 7, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 10 vs 10

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -3 y1 - 3 y2 - 3 y3 - 6 y4 + 10 y6 - 3 y5, 3 y1, 0, 3 y2, 6 y4, 3 y3, -3 y4 + 3 y6, 3 y4, 3 y5, 3 y6]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]


 
1395 . Coloring, {3, 6, 8, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, B, B, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y7, 2 y6, y6, y8, 0]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y5, 0, y7, 0, y6, y3, y4, y5, 0, y1, 0, y2]

  p = - s 3 + s 8


 
1396 . Coloring, {3, 6, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, B, C, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   10

See Matrix
 

[0, y1, y1 - y2 + y3 - y4 + y5 + y6 - y7 + y8 - y9, y2, 0, y3, y4, y5, y6, y7, y8, y9]

  p = - s 9 + s 10

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y2, 0, y1, y9, y8]


 
1397 . Coloring, {3, 7, 8, 9, 10, 11}

R: [7, 7, 8, 6, A, A, A, B, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, y7 + y4, y5, y6, y7, 0, y3, y4, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y4, 0, y1, 0, 0, y2, y3, y5, y6, 0, y8, y7]


 
1398 . Coloring, {3, 7, 8, 9, 10, 12}

R: [7, 7, 8, 6, A, A, A, B, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y7 - 3 y6 + 13 y5 - 3 y4 - 3 y2 + 13 y3, 0, 0, 0, 3 y7, 3 y6, 3 y7, 3 y5, 3 y4, 3 y2, 3 y3]

  p = - s 4 - s 5 + s 7 + s 8   p' = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y7, y6, y5, y4, y3, y4, 0, 0, y2, y1]

  p = s 5 - s 8


 
1399 . Coloring, {3, 7, 8, 9, 11, 12}

R: [7, 7, 8, 6, A, A, A, B, C, C, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y5, 2 y6, y6, y7, y4, y2, y3]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, 0, y8, y9]


 
1400 . Coloring, {3, 7, 8, 10, 11, 12}

R: [7, 7, 8, 6, A, A, A, B, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, 0, y4, 0, y3, y1, y2, 2 y2, y6, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y8, y6, y7, y5, 0, 0, y4, y3]


 
1401 . Coloring, {3, 7, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, A, A, C, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, y1, 0, 10 y1 + 10 y2 + 10 y3 - 22 y4 + 10 y5 - 22 y6, 0, y2, y3, 5 y1 + 5 y2 + 5 y3 - 11 y4 + 5 y5 - 11 y6, y4, y5, 0, y6]

  p = s 3 + s 4 - s 6 - s 7   p' = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y5, y6, y4, y3, 0, 0, y7, 2 y3]

  p = - s 3 + s 8


 
1402 . Coloring, {3, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, A, B, B, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y7, 0, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y6 - 3 y4 - 3 y3 + 8 y2 + 5 y1, -3 y7 - 3 y5 + 5 y2 + 8 y1, 3 y1]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y7, 0, y6, y7, y8]

  p = - s 4 + s 9


 
1403 . Coloring, {4, 5, 6, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, C, 1, 5]
B: [6, 8, 8, 6, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, 0, y1 + y2 - y5 - y3 - y4 + y7 + y6, 0, y2, 0, y5, y3, y4, y7, y6]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1404 . Coloring, {4, 5, 6, 7, 8, 10}

Ωp(Δ)=0:     p = 3s 2 - 14s 3 + 24s 5 + 32s 6 + 32s 7 - 128s 8   p' = - 3s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, A, B, B, 2, 1, 5]
B: [6, 8, 8, 6, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y1, y2, y1, 0, y5, 0, y3, 0, 0, y4, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, 0, y5, 0, y3, y1, y2, y3, y4 - y5 + y1 + y2]

  p' = - s 5 + s 6   p = s 5 - s 6


 
1405 . Coloring, {4, 5, 6, 7, 8, 11}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7   p = 3s 2 + 14s 3 - 24s 5 + 32s 6 - 32s 7 - 128s 8

R: [7, 7, 7, 7, 3, 3, A, B, B, C, 4, 5]
B: [6, 8, 8, 6, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 + y2 - y3 - y5 - y6 + y4 + y7, y1, 0, 0, 0, y2, 0, y3, y5, y6, y4, y7]

  p = - s 7 + s 8


 
1406 . Coloring, {4, 5, 6, 7, 8, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 - 24s 5 + 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, A, B, B, C, 1, 9]
B: [6, 8, 8, 6, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, 0, y4, 0, y3, y2, y1, y6 + y5 - y4 + y3 + y2 - y1]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y3, 0, y1, y2, y6, 0, y7, 0, y4, y5, y8]


 
1407 . Coloring, {4, 5, 6, 7, 9, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, 3, A, C, C, 2, 1, 5]
B: [6, 8, 8, 6, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y1, y2, 0, y3, 0, y4, 0, 0, y5, 0, y6]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2 - y1 - y3 - y4 + y6 + y5, 0, y2, 0, y1, y3, y4, y6, y5]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1408 . Coloring, {4, 5, 6, 7, 9, 11}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, A, C, C, C, 4, 5]
B: [6, 8, 8, 6, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y1, y6, 0, y5, 0, 0, y4, 0, y3]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y4 - y3 - y5 + y6, y1, 0, 0, 0, y2, 0, y4, y3, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1409 . Coloring, {4, 5, 6, 7, 9, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, A, C, C, C, 1, 9]
B: [6, 8, 8, 6, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 6 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y1, 0, y1, 0, 0, 0, 2 y1 + y2 + y3 - y4, 0, y2, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, 0, y4, y2, y1, 0, y7, 0, y6, y5, 0]


 
1410 . Coloring, {4, 5, 6, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, 2, 4, 5]
B: [6, 8, 8, 6, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, 0, y4, 0, 0, y5, y6, y6]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y1 + y2 + y3 + y4 - y5 - y6, 0, y2, y3, y4, y5, y6]

  p = - s 6 + s 7


 
1411 . Coloring, {4, 5, 6, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, 2, 1, 9]
B: [6, 8, 8, 6, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 2 y7, 0, 0, 0, y3, 0, y6, y4, y5, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, 0, y4, 0, y5, y6, y7]


 
1412 . Coloring, {4, 5, 6, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, C, 4, 9]
B: [6, 8, 8, 6, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y3 + y1 - y2 - y5 + y6 + y4, y3, 0, 0, y1, 0, y2, y5, y6, y4]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, y3, y6, 0, y4, 0, y5, y8, y7]


 
1413 . Coloring, {4, 5, 6, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, 2, 1, 5]
B: [6, 8, 8, 6, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y4, 2 y5, y3, 0, y2, 0, y1, 0, 0, 0, y6, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, 0, y3, y6, y7, y4, y5]


 
1414 . Coloring, {4, 5, 6, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, C, 4, 5]
B: [6, 8, 8, 6, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 6 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y3, y2, 0, y4, 0, 0, 0, y6, y5]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y3, 0, y4, y5, y6, y7, y8]


 
1415 . Coloring, {4, 5, 6, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, C, 1, 9]
B: [6, 8, 8, 6, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y1, 0, -5 y1 - 5 y5 + 11 y4 - 5 y2 + 11 y3, 0, 0, 0, 5 y5, 0, 5 y4, 0, 5 y2, 5 y3]

  p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y2, 0, y1, y3, y8, 0, y4, 0, y6, y7, y5]


 
1416 . Coloring, {4, 5, 6, 8, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, B, B, B, 2, 4, 5]
B: [6, 8, 8, 6, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 5 vs 6

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y3, y1, y2, y3, 0, y4, 0, 0, 0, y5, 0]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y2, 0, 0, 0, 0, y1, 0, y2, y4, y5, 0, y3]

  p = s 4 - s 6


 
1417 . Coloring, {4, 5, 6, 8, 10, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, 3, B, B, B, 2, 1, 9]
B: [6, 8, 8, 6, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y4, y4, 0, 0, 0, y2, 0, y4, 0, y3, 0]

  p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, y3, y1, y2, 0, y3, 0, y4, 0, y5]

  p = - s 3 + s 6


 
1418 . Coloring, {4, 5, 6, 8, 11, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, B, B, B, C, 4, 9]
B: [6, 8, 8, 6, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, y3, y2, 0, 0, y1, 0, y5, 0, y4, y3]

  p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, y4, 0, y5, 0, y6, 0, y7]


 
1419 . Coloring, {4, 5, 6, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, B, C, C, 2, 4, 5]
B: [6, 8, 8, 6, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5, y4, y3, y2, 0, y1, 0, 0, 0, y6, y5]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y5, 0, y2, y3, y4, y6, y7]


 
1420 . Coloring, {4, 5, 6, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, B, C, C, 2, 1, 9]
B: [6, 8, 8, 6, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-2 y1 - y3 + 6 y4 - y2, y1, y1, 0, 0, 0, y3, 0, y4, 0, y2, y4]

  p' = s 3 - s 6   p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y7, y6, 0, y5, 0, y4, y3, y2]


 
1421 . Coloring, {4, 5, 6, 9, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, B, C, C, C, 4, 9]
B: [6, 8, 8, 6, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, -3 y1 - 3 y2 + 5 y3 - 3 y4 + 5 y5, 3 y1, 0, 0, 3 y2, 0, 3 y3, 0, 3 y4, 3 y5]

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5 + y6, y1, 0, 0, y2, y3, 0, y4, 0, y5, y6, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1422 . Coloring, {4, 5, 6, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, C, B, 2, 4, 9]
B: [6, 8, 8, 6, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y5, 2 y5, y1, 0, 0, y2, 0, y3, 0, y4, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y3, y4, 0, y2, 0, y6, y7, y5]


 
1423 . Coloring, {4, 5, 7, 8, 9, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 + 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, A, A, B, C, 2, 1, 5]
B: [6, 8, 8, 6, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y4, y3, y2, 0, y4, 0, y1, 0, 0, y6, y5, y5]

  p = - s 4 + s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y2, 0, y3, 0, y4, y5, y6, y7, y8]


 
1424 . Coloring, {4, 5, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, A, A, B, C, C, 4, 5]
B: [6, 8, 8, 6, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, y2, y3, 0, y1, 0, 0, y6, y7, y5]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, y2, y3, 0, 0, y7, 0, y4, y5, y6, y8, y9]


 
1425 . Coloring, {4, 5, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, A, B, C, C, 1, 9]
B: [6, 8, 8, 6, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y5 - y4 - y3 + y2, 0, y1, 0, 0, 0, y5, 0, y4, y3, y1, y2]

  p = - s 5 + s 7   p = - s 5 + s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y6, y7, y8, y4, 0, y5, 0, y2, y3, y9]


 
1426 . Coloring, {4, 5, 7, 8, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y2, y3, -2 y2 + 2 y3, 0, y4, 0, 0, y5, -2 y2 + 2 y3, 0]

  p' = s 3 - s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y4, y6, y5, 2 y5, y7]

  p = - s 6 + s 8


 
1427 . Coloring, {4, 5, 7, 8, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, 0, 0, y4, 0, 2 y3, y5, y6, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, 0, y5, 0, y6, y7, y8]


 
1428 . Coloring, {4, 5, 7, 8, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, C, 4, 9]
B: [6, 8, 8, 6, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 - y3 - y4 + y5 + y6, y1, 0, 0, y2, 0, y3, y4, y5, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, 0, y6, 0, y7, y8, y9]


 
1429 . Coloring, {4, 5, 7, 9, 10, 11}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, y1, y5, y2, 0, y3, 0, 0, y4, 0, y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y4, y5, y6, y8, y7]


 
1430 . Coloring, {4, 5, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[2 y1, -3 y1 - y4 - y2 + 6 y3, y1, 0, 0, 0, y4, 0, y3, y2, 0, y3]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 5 y7, 5 y5, 5 y6, 5 y4, 0, 5 y3, 0, 5 y2, -5 y7 - 5 y5 + 11 y6 - 5 y4 - 5 y3 + 11 y2 + 11 y1, 5 y1]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1431 . Coloring, {4, 5, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, C, 4, 9]
B: [6, 8, 8, 6, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 6 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, 0, 3 y1 + y3 + y2 - y4, 0, y3, y2, 0, y4]

  p = s 4 - s 6   p' = s 4 - s 5

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, 0, y7, 0, y8, y6, 0]


 
1432 . Coloring, {4, 5, 7, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, A, A, C, B, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y7, y1, 0, 0, y3, 0, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[3 y7, 0, 3 y6, 0, 3 y5, 3 y4, 0, 3 y3, 0, 3 y2, -3 y7 - 3 y6 + 5 y5 - 3 y4 - 3 y3 + 5 y2 + 5 y1, 3 y1]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1433 . Coloring, {4, 5, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, B, C, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5, y4, y3, y5, 0, y2, 0, 0, y6, y1, y6]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y4, y5, y6, y7, y8]


 
1434 . Coloring, {4, 5, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 6s 3 + 40s 5 - 32s 6 - 32s 7 + 128s 8   p' = - s 2 + 2s 3 + 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, B, C, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y4 - 3 y2 + 13 y5 - 3 y3 + 13 y6, 3 y4, 0, 0, 0, 3 y2, 0, 3 y5, 3 y4, 3 y3, 3 y6]

  p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y5, y4, y3, y2, 0, y1, 0, y8, y7, y6]


 
1435 . Coloring, {4, 5, 8, 9, 11, 12}

Ωp(Δ)=0:     p' = - s 2 - 2s 3 + 8s 4 + 8s 5 - 32s 7   p = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, B, B, C, C, 4, 9]
B: [6, 8, 8, 6, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 3 y2, 3 y1, 0, 0, -3 y2 - 3 y1 + 10 y4 - 3 y3, 0, -3 y2 + 3 y4, 3 y2, 3 y3, 3 y4]

  p' = - s 3 + s 6   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y9, y8, y7, 0, y6, y5, 0, y4, 0, y3, y2, y1]


 
1436 . Coloring, {4, 5, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, B, B, B, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1, y5, y2, 0, 0, y3, 0, 2 y5, y5, y4, 0]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, 0, y7]


 
1437 . Coloring, {4, 5, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, B, C, C, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, y1, y3, -y1 - 2 y3 - y2 - y5 + 6 y4, 0, 0, y2, 0, y4, y3, y5, y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[7 y7, 0, 7 y6, 0, 7 y5, 7 y4, 0, 7 y3, 0, 7 y2, -7 y7 - 7 y6 + 9 y5 - 7 y4 - 7 y3 + 9 y2 + 9 y1, 7 y1]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1438 . Coloring, {4, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = 3s 2 - 2s 3 - 8s 5 - 32s 7   p' = - 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, A, B, C, 2, 1, 5]
B: [6, 8, 8, 6, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y1, y2, y5, 0, y1, 0, y3, 0, 0, y4, y5, y5]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y2, y1, 0, y3, 0, y5, y4, y6, y7, y8]


 
1439 . Coloring, {4, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 9s 2 - 4s 4 + 24s 5 - 16s 6 + 96s 7 - 64s 8   p' = 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, A, B, C, C, 4, 5]
B: [6, 8, 8, 6, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y5, y1, y2, 0, y3, 0, 0, y4, y5, y6]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, 0, y5, y6, y7, y8, y9]


 
1440 . Coloring, {4, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7   p = 3s 2 + 14s 3 - 24s 5 + 32s 6 - 32s 7 - 128s 8

R: [7, 7, 7, 7, A, 3, A, B, C, C, 1, 9]
B: [6, 8, 8, 6, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y5 - y4 - y3 + y1, 0, y2, 0, 0, 0, y5, 0, y4, y3, y2, y1]

  p = - s 5 + s 7   p = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y6, y7, y8, y9, 0, y5, 0, y2, y3, y4]


 
1441 . Coloring, {4, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y2, y3, 2 y2, 0, y4, 0, 0, y5, 2 y2, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-2 y3 + 2 y2, 0, y3, 0, 0, y1, 0, y2, y4, y5, 2 y3, y6]

  p = - s 5 + s 7   p' = - s 5 + s 7


 
1442 . Coloring, {4, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, 0, y5, 0, 2 y3, y6, y4, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y4, y2, y3, 0, y8, 0, y7, y5, y6]


 
1443 . Coloring, {4, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = - s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, C, 4, 9]
B: [6, 8, 8, 6, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y4, 0, 0, y2, 0, -y3 - y4 + y2 - y1 + y6 + y5, y1, y6, y5]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y3, y2, y1, 0, y5, y4, 0, y8, 0, y9, y7, y6]


 
1444 . Coloring, {4, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y2, y1, 2 y1, y3, 0, y4, 0, 0, y5, 0, 2 y1]

  p = s 3 - s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, y5, 0, y4, y3, y6, y7, y8]


 
1445 . Coloring, {4, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[2 y3, y4, y3, 0, 0, 0, y2, 0, y1, -y4 - 3 y3 - y2 + 6 y1, 0, y1]

  p = - s 2 + s 5   p' = s 3 - s 6   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4, y3, y1, y2, 0, y8, 0, y7, y5, y6]


 
1446 . Coloring, {4, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, C, 4, 9]
B: [6, 8, 8, 6, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 6 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, 0, 3 y1 + y2 + y3 - y4, 0, y2, y3, 0, y4]

  p = s 4 - s 5   p' = - s 4 + s 5

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y4, y3, y2, 0, y1, y6, 0, y5, 0, y8, y7, 0]


 
1447 . Coloring, {4, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, A, C, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y7, y2, 0, 0, y3, 0, y4, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, y7, y8]


 
1448 . Coloring, {4, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, A, 3, B, B, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y7, y2, y3, 0, y4, 0, 0, y5, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y3, 0, y6, y4, y5, y8, y7]


 
1449 . Coloring, {4, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, B, B, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 6 y4 - 3 y2 + 13 y3 - 3 y5 + 13 y6, 3 y1, 3 y4, 0, 0, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y4, 0, y5, 0, y6, y7, y8]


 
1450 . Coloring, {4, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, B, B, C, C, 4, 9]
B: [6, 8, 8, 6, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -3 y2 + 3 y4, 3 y2 + 7 y4 - 3 y1 - 3 y3, 0, 0, 3 y1, 0, 3 y2, -3 y2 + 3 y4, 3 y3, 3 y4]

  p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y4, y3, y1, 0, y2, y9, 0, y7, 0, y8, y6, y5]


 
1451 . Coloring, {4, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, B, B, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1, y4, y2, 0, 0, y3, 0, 2 y4, y4, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, 0, y7]


 
1452 . Coloring, {4, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, B, C, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -2 y3 - y1 - y2 + 6 y5 - y4, y3, y1, 0, 0, y2, 0, y5, y3, y4, y5]

  p' = - s 4 + s 7   p' = s 3 - s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, y7, y8]


 
1453 . Coloring, {4, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y3, 0, y2, y2, 0, y1, 0, 0, y5, y4, y4]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y6, 0, y7, 0, 0, y1, 0, y2, y3, 0, y4, y5]


 
1454 . Coloring, {4, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 + 13 y4 - 3 y5 - 3 y6 + 13 y3, 3 y1, 0, 0, 0, 0, 3 y2, 0, 3 y4, 3 y5, 3 y6, 3 y3]

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y3, y1, y2, y4, 0, y5, 0, 0, y6, y7]


 
1455 . Coloring, {4, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, C, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3 - y1 - y2 + y5 + y4, 0, 0, y3, 0, y1, y2, y5, y4]

  p = - s 5 + s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y7, 2 y6, 2 y5, 0, 2 y3, 2 y4, 0, 2 y2, 0, 0, 3 y6, 2 y1]

  p = - s 4 + s 8


 
1456 . Coloring, {4, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, B, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 6 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, 0, y4, y5, y6, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[-y1 + y2 + y5 + y3 + y4 - y6, 0, y1, 0, y2, y5, 0, y3, 0, 0, y4, y6]

  p = s 4 - s 5 + s 6 - s 7


 
1457 . Coloring, {4, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, A, A, C, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 2 vs 6 7 vs 7

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -y2 + 2 y1, 0, y2, 0, 0, 2 y1, 0, y1, 2 y1, 0, y1]

  p' = s 2 - s 5   p' = s 3 - s 5   p' = s 4 - s 5   p = s 2 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, 0, y3, 0, 0, y2, y1]


 
1458 . Coloring, {4, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, B, B, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 + 13 y3 - 3 y4 - 3 y5 + 13 y6, 0, 3 y1, 0, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y5, 0, y4, 0, y3, y2, 0, y1, 0, 2 y7, y7, y6]

  p = - s 4 + s 8


 
1459 . Coloring, {5, 6, 7, 8, 9, 10}

R: [7, 7, 7, 6, 3, 3, A, B, C, 2, 1, 5]
B: [6, 8, 8, 7, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, y2, y6, y4, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y4, 0, y3, y1, 2 y3, y2, y5, y6, y7]

  p = - s 5 + s 8


 
1460 . Coloring, {5, 6, 7, 8, 9, 11}

R: [7, 7, 7, 6, 3, 3, A, B, C, C, 4, 5]
B: [6, 8, 8, 7, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y1, y4, y3, y5, 0, 0, y7, y6, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y2, y3, 0, 0, 0, y1, y4, y5, y6, y7, y8, y9]


 
1461 . Coloring, {5, 6, 7, 8, 9, 12}

R: [7, 7, 7, 6, 3, 3, A, B, C, C, 1, 9]
B: [6, 8, 8, 7, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y2, 0, y2, 0, 0, y4, y3, 0, -2 y2 + 2 y4 + y3 - y1 + y5, y1, y4, y5]

  p = s 5 - s 6   p' = - s 5 + s 6   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, 5 y1 - 3 y2 - 3 y3 + 5 y4 - 3 y5 - 3 y6 + 5 y7 - 3 y8, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 0, 3 y6, 3 y7, 3 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1462 . Coloring, {5, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = 9s 3 - 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, 2, 4, 5]
B: [6, 8, 8, 7, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y7, y4, y5, 0, 0, y6, y7, 0]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y4, y3, 2 y3, y2, y7, y5, y6]

  p = s 6 - s 8


 
1463 . Coloring, {5, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = 6s 2 - 5s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, 2, 1, 9]
B: [6, 8, 8, 7, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y3, y1, 0, 0, y5, y4, 0, 2 y5, y6, 2 y1 - 2 y5, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, -4 y3 - 2 y1 + 2 y2 + 2 y4, 16 y3 + 9 y1 - 11 y2 - 2 y4, 2 y3, 14 y3 + 7 y1 - 9 y2 - 2 y4, 4 y3, 0, 2 y1, 2 y2, 2 y4]

  p = s 2 - s 8   p' = s 2 - s 5   p' = s 3 - s 6   p' = s 4 - s 7


 
1464 . Coloring, {5, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, C, 4, 9]
B: [6, 8, 8, 7, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2, y3, 0, 0, y1, y4, y5, y6, 0, y7, y9, y8]


 
1465 . Coloring, {5, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, 2, 4, 5]
B: [6, 8, 8, 7, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, 2 y3 - 2 y4, y3, y4, y5, 0, 0, y6, 0, 2 y3 - 2 y4]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y6, y7, 2 y7, y4, y5, y3, y2]

  p = - s 2 + s 8

 ยป SYNC'D !RANK'D


 
1466 . Coloring, {5, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, 2, 1, 9]
B: [6, 8, 8, 7, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 5 vs 8 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[2 y1, -y2 - 3 y1 - y5 + 6 y3 - y4, y2, 0, 0, y1, y5, 0, y3, y4, 0, y3]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y3, y4, -10 y3 - 7 y4 + 16 y1 + 3 y2, y2, -20 y3 - 14 y4 + 32 y1 + 6 y2, 0, y1, -8 y3 - 4 y4 + 2 y2 + 13 y1, -11 y3 + 18 y1 - 8 y4 + 4 y2]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8

 ยป SYNC'D !RANK'D


 
1467 . Coloring, {5, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = - 2s 2 + s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, C, 4, 9]
B: [6, 8, 8, 7, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6, y1, 0, y2, y3, 0, y4, y5, 0, y6]

  p = - s 6 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y6, y5, 0, 0, 2 y3, y4, y3, y2, 0, y1, y7, 0]

  p = - s 2 + s 8


 
1468 . Coloring, {5, 6, 7, 10, 11, 12}

R: [7, 7, 7, 6, 3, 3, A, C, B, 2, 4, 9]
B: [6, 8, 8, 7, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y9, y8, y7, 0, y6, y5, 0, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y3, 0, 0, 0, y2, y1, y7, 2 y7, 0, y5, y6, y4]

  p = - s 5 + s 8


 
1469 . Coloring, {5, 6, 8, 9, 10, 11}

R: [7, 7, 7, 6, 3, 3, B, B, C, 2, 4, 5]
B: [6, 8, 8, 7, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y7, y4, y1, y2, y3, y5, 0, 0, 0, y6, y7]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y3, y2, 2 y2, y4, y7, y6, y5]

  p = - s 2 + s 8


 
1470 . Coloring, {5, 6, 8, 9, 10, 12}

R: [7, 7, 7, 6, 3, 3, B, B, C, 2, 1, 9]
B: [6, 8, 8, 7, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 9 y2 - 3 y4 + 13 y5 - 3 y3 + 13 y6, 6 y2, 3 y1, 0, 0, 3 y2, 3 y4, 0, 3 y5, 0, 3 y3, 3 y6]

  p = s 3 + s 4 - s 6 - s 7   p' = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y5, y4, 2 y5, 0, y3, y5, y6]

  p' = s 4 - s 7   p = - s 4 + s 7


 
1471 . Coloring, {5, 6, 8, 9, 11, 12}

R: [7, 7, 7, 6, 3, 3, B, B, C, C, 4, 9]
B: [6, 8, 8, 7, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -5 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 + 11 y6, 5 y1, 0, 5 y2, 5 y3, 0, 5 y4, 0, 5 y5, 5 y6]

  p = - s - s 2 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2, y1, 0, 0, y8, y7, y6, y5, 0, y4, y6, y3]

  p = - s 4 + s 9


 
1472 . Coloring, {5, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, B, B, 2, 4, 9]
B: [6, 8, 8, 7, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y5, y4, 0, y3, y2, 0, y6, 0, y1, 0]

  p = s 2 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y5, 0, 0, 0, y1, y2, y5, 2 y5, 0, y3, 0, y4]

  p = - s 3 + s 6   p' = - s 3 + s 6

 ยป SYNC'D !RANK'D


 
1473 . Coloring, {5, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 5s 3 - 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, B, C, C, 2, 4, 9]
B: [6, 8, 8, 7, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, y6, y5, y4, 0, y3, y2, 0, y1, 0, -y6 - y5 - y4 - y3 - y2 + 6 y1, y1]

  p' = - s 2 + s 7   p = - s 2 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y2, y3, 2 y3, 0, y6, y7, y5]

  p = - s 5 + s 8


 
1474 . Coloring, {5, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = - 2s 2 + 5s 3 + 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, 2, 4, 5]
B: [6, 8, 8, 7, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, y5, y4, y4, y5, y3, 0, 0, y2, y1, y1]

  p' = s 4 - s 7   p = s 4 - s 7   p' = s 5 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y2, 0, y1, 0, 0, y4, y6, y3, y5, y6, y7, y8]

  p = - s 2 + s 9

 ยป SYNC'D !RANK'D


 
1475 . Coloring, {5, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, 2, 1, 9]
B: [6, 8, 8, 7, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y5, 3 y4, 3 y3, 0, 0, 3 y3, 3 y1, 0, 3 y2, -3 y5 - 3 y4 - 9 y3 - 3 y1 + 13 y2 + 13 y6, 3 y3, 3 y6]

  p' = - s 3 - s 4 + s 6 + s 7   p = s 3 + s 4 - s 6 - s 7   p' = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, -6 y1 + 5 y2 - 6 y3 + 4 y4 + 4 y6 - y5, 5 y1 - 4 y2 + 5 y3 - 2 y4 - 2 y6, y4, 0, y6, -8 y1 + 6 y2 - 8 y3 + 5 y4 + 5 y6, y5]

  p' = s 5 - s 8   p = - s 4 + s 7   p' = - s 4 + s 7


 
1476 . Coloring, {5, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, C, 4, 9]
B: [6, 8, 8, 7, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1 - y4, 0, y2 + y3 - y4 - y5, y1, 0, y2, y3, y4, y5]

  p' = - s 5 + s 7   p' = - s 5 + s 6   p = s 5 - s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[y5, y6, y7, 0, y1, y2, y3, y4, 0, y8, y9, y10]


 
1477 . Coloring, {5, 7, 8, 10, 11, 12}

R: [7, 7, 7, 6, 3, A, A, B, B, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y3, y2, 0, y4, y7, 0, 2 y3, y6, y5, 0]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
1478 . Coloring, {5, 7, 9, 10, 11, 12}

R: [7, 7, 7, 6, 3, A, A, C, C, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - y2 - y3 - y5 + 6 y4, y1, 2 y1, 0, y2, y3, 0, y4, y5, 0, y4]

  p = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[5 y1, 0, -5 y1 + 11 y2 - 5 y3 - 5 y4 - 5 y5 + 11 y6 - 5 y7 + 11 y8, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1479 . Coloring, {5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, B, B, C, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y4 + 8 y2 - 3 y6 + 5 y7, 3 y1, -3 y5 + 5 y2 - 3 y3 + 8 y7, 0, 3 y4, 3 y5, 0, 3 y2, 3 y3, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y3, 0, y2, 0, y1, y8, y7, y6, 0, y5, y7, y4]

  p = s 6 - s 9

 ยป SYNC'D !RANK'D


 
1480 . Coloring, {6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = - 6s 2 - s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, 2, 4, 5]
B: [6, 8, 8, 7, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, y6, y4, 0, 0, y5, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y5 - y4 - y3 - y1 + y2 + y7, 0, y6, 0, 0, y5, y6, y4, y3, y1, y2, y7]

  p = s 3 - s 9   p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1481 . Coloring, {6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, 2, 1, 9]
B: [6, 8, 8, 7, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y1 + 3 y4, -6 y1 - 9 y4 - 3 y5 + 13 y6 - 3 y3 + 13 y2, 3 y1, 0, 0, 3 y4, 3 y5, 0, 3 y6, 3 y3, 3 y4, 3 y2]

  p = s 3 - s 9   p' = - s 3 + s 5 + s 6 - s 8   p' = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, 5 y7, 5 y6, 5 y5, 5 y4, 5 y3, 5 y2, 0, 5 y1, -5 y6 + 6 y5 - 5 y3 + 6 y2 + 6 y1, -5 y7 + 5 y5 - 5 y4 + 5 y2 + 5 y1]

  p = - s 3 - s 4 - s 5 + s 7 + s 8 + s 9   p = - s 3 - s 5 + s 6 + s 8


 
1482 . Coloring, {6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, C, 4, 9]
B: [6, 8, 8, 7, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 - y5 + y6 + y7, y1, 0, y2, y3, 0, y4, y5, y6, y7]

  p = s 7 - s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1 + y2 - y3 + y4 - y5 - y6 - y7 + y8 + y9, y1, y2, 0, y3, y4, y5, y6, 0, y7, y8, y9]

  p = - s 7 + s 8 - s 9 + s 10


 
1483 . Coloring, {6, 7, 8, 10, 11, 12}

R: [7, 7, 7, 6, A, 3, A, B, B, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y6, 0, y7, y3, 0, y4, y5, y8, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y4 - y8 - y5 + y6 + y7, 0, y1, 0, y2, y3, y4, y8, 0, y5, y6, y7]

  p = s 6 - s 7 + s 8 - s 9


 
1484 . Coloring, {6, 7, 9, 10, 11, 12}

R: [7, 7, 7, 6, A, 3, A, C, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -y1 - y2 - y6 - y3 + 6 y5 - y4, y1, y2, 0, y6, y3, 0, y5, y4, 0, y5]

  p' = s 4 - s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y8 - y4 - y5 + y6 + y7, 0, y1, 0, y2, y3, y8, y4, 0, y5, y6, y7]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1485 . Coloring, {6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, A, 3, B, B, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -3 y4 - 3 y3 - 3 y2 - 3 y1 + 13 y8 - 3 y7 - 3 y6 + 13 y5, 3 y4, 3 y3, 0, 3 y2, 3 y1, 0, 3 y8, 3 y7, 3 y6, 3 y5]

  p = s 3 + s 4 - s 8 - s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y2, 0, y1, 0, y8, y3, y6, y4, 0, y5, y6, y7]

  p = - s 5 + s 9


 
1486 . Coloring, {7, 8, 9, 10, 11, 12}

R: [7, 7, 7, 6, A, A, A, B, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y3 - 3 y2 + 13 y6 - 3 y4 - 3 y5 + 13 y7, 0, 3 y1, 0, 3 y3, 3 y2, 0, 3 y6, 3 y4, 3 y5, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, y3, y2, -y5 - y4 + y3 + y2 + y1 + y6 - y7, y1, 0, 0, y6, y7]

  p = - s 5 + s 6 - s 7 + s 8


 
1487 . Coloring, {2, 3, 4, 5, 6, 7, 8}

Ωp(Δ)=0:     p' = s 6   p' = s 2   p' = s 3   p' = s 4   p = s 2   p' = s 5   p' = s 7

R: [7, 8, 8, 7, 3, 3, A, B, B, C, 1, 5]
B: [6, 7, 7, 6, A, A, B, C, C, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 8 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y1, 0, y1, 0, y1, y1, 0, y1, y1, y1]

  p = - s + s 2   p = - s + s 3   p = - s + s 4   p = - s + s 6   p = - s + s 5   p = - s + s 7   p = - s + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, y1, 0, y1, 0, y1, y1, 0, y1, y1, y1, y1]

  p = - s + s 4   p = - s + s 6   p = - s + s 5   p = - s + s 3   p = - s + s 2   p = - s + s 7   p = - s + s 8


` See 8-level graph

`

M              \ ;       N

$ [ [0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1] , [0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1] , [1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1] , [0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1] , [1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1] , [0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2] , [1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1] , [0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0] ] $     $ [ [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0] ] $

τ= 20 , r'= 7/8

R: [7, 8, 8, 7, 3, 3, A, B, B, C, 1, 5]
B: [6, 7, 7, 6, A, A, B, C, C, 2, 4, 9]

Ranges

Action of R on ranges, [[1], [1]]
Action of B on ranges, [[2], [2]]

Cycles:    R , {{1, 3, 5, 7, 8, 10, 11, 12}},   B , {{2, 4, 6, 7, 10, 11}, {9, 12}}

β({1, 3, 5, 7, 8, 10, 11, 12}) = 1/2
β({2, 4, 6, 7, 9, 10, 11, 12}) = 1/2

Partitions
α([{12}, {10}, {11}, {2, 3}, {5, 6}, {1, 4}, {8, 9}, {7}]) = 1/1

b1 = {12} ` , ` b2 = {10} ` , ` b3 = {11} ` , ` b4 = {2, 3} ` , ` b5 = {5, 6} ` , ` b6 = {1, 4} ` , ` b7 = {8, 9} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [2, 8, 7, 5, 1, 3, 4, 6] [7, 5, 8, 2, 6, 3, 1, 4]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-8 partition graph.

`

Right Group
Coloring {2, 3, 4, 5, 6, 7, 8}
Rank8
R,B [7, 8, 8, 7, 3, 3, A, B, B, C, 1, 5], [6, 7, 7, 6, A, A, B, C, C, 2, 4, 9]
π2 [0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2]
u2 [1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 1)
wpp [2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1]
π8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

 

 
1488 . Coloring, {2, 3, 4, 5, 6, 7, 9}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, A, C, C, C, 1, 5]
B: [6, 7, 7, 6, A, A, B, B, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y1, 0, -y2 + y3, 0, -y1 + y3, 0, y2, y3 - y4, 0, y4, 0, y3 ]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y4, 0, y3, 0, y2, y1, 0, y4 - y3 + y2 - y1 - y6 + y5, y6, y5, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1489 . Coloring, {2, 3, 4, 5, 6, 7, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, 2, 1, 5]
B: [6, 7, 7, 6, A, A, B, B, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1 + y2 - y3 + y4 - y5 - y6 + y7 + y8, y1, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

  p = s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - y5 - y3 - y2 + y4 + y6, 0, y1, y5, 0, y3, y2, y4, y6]

  p = - s 6 + s 7


 
1490 . Coloring, {2, 3, 4, 5, 6, 7, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, C, 4, 5]
B: [6, 7, 7, 6, A, A, B, B, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y4 + y5 - y2, -y1 + y4 + y5, y1, 0, y2, -y3 + y4 + y5, 0, y3, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[-3 y2 + 5 y3 - 3 y4 + 8 y6, -3 y1 + 8 y3 - 3 y5 + 5 y6, 0, 0, 0, 3 y1, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = s - s 7   p' = s - s 7


 
1491 . Coloring, {2, 3, 4, 5, 6, 7, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, C, 1, 9]
B: [6, 7, 7, 6, A, A, B, B, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, y6, 0, 0, 0, y7 - y6 + y1 + y2 + y3 - y4 - y5, y1, y2, y3, y4, y5]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y3, y4, y5, y2, 0, 0, y6, y7, y8]


 
1492 . Coloring, {2, 3, 4, 5, 6, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, B, B, C, C, 1, 5]
B: [6, 7, 7, 6, A, A, A, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, 0, y6, 0, y5, 0, y3, y4, 0, 0, y2, y1]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y4, 0, y3, y2, 0, y7, y6, y8, y5]


 
1493 . Coloring, {2, 3, 4, 5, 6, 8, 10}

Ωp(Δ)=0:     p = s 2 + 56s 5 + 16s 6 - 64s 7 - 256s 8   p' = s 2 - 16s 4 - 8s 5 + 16s 6 + 64s 7   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, B, 2, 1, 5]
B: [6, 7, 7, 6, A, A, A, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 6 vs 7 4 vs 6

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y5, y4, y3, 0, y4, 0, y2, y1, 0, 0, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, y3, 0, y4, y3, 0, y2, y1, 0, -y4 + y2 + y1]

  p' = s 4 - s 5   p = s 4 - s 6


 
1494 . Coloring, {2, 3, 4, 5, 6, 8, 11}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, 3, 3, B, B, B, C, 4, 5]
B: [6, 7, 7, 6, A, A, A, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 3 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y7, 0, 0, y5, y6]

Omega Rank for B :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-y1 + 2 y3, y1, 0, 0, 0, -y2 + 2 y3, y2, 0, y3, 2 y3, 0, y3]

  p = - s 3 + s 4   p = - s 3 + s 5   p = - s 3 + s 6   p = - s 3 + s 7


 
1495 . Coloring, {2, 3, 4, 5, 6, 8, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, B, B, B, C, 1, 9]
B: [6, 7, 7, 6, A, A, A, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, 0, y2, 0, 0, 0, y5, y4, y4, 0, y3, y2]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, y5, y2, y2, y3, 0, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1496 . Coloring, {2, 3, 4, 5, 6, 9, 10}

Ωp(Δ)=0:     p = s 2 - 8s 5 - 80s 6 - 64s 7 - 128s 8   p' = s 2 - 8s 4 - 8s 5 - 16s 6   p' = s 3 - 8s 5 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, C, C, 2, 1, 5]
B: [6, 7, 7, 6, A, A, A, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[y4, -y1 - y2 + 5 y4 - y3 - y5, y1, 0, y2, 0, y4, y3, 0, 0, y4, y5]

  p = - s 2 + s 6   p' = - s 2 + s 6   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, 0, y4, y5, 0, y1, y2, y3, y6 - y4 - y5 + y1 + y2 - y3]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1497 . Coloring, {2, 3, 4, 5, 6, 9, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, 3, B, C, C, C, 4, 5]
B: [6, 7, 7, 6, A, A, A, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 0, y2, y3, y1, 0, y3, -y2 - y1 + 5 y3 - y4, 0, 0, y3, y4]

  p' = s 2 - s 6   p' = s - s 5   p = s - s 5

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, 0, y5, y6, y7, 0]


 
1498 . Coloring, {2, 3, 4, 5, 6, 9, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, 3, 3, B, C, C, C, 1, 9]
B: [6, 7, 7, 6, A, A, A, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 6 vs 7

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[y3, 0, 5 y3 - y1 - y2 - y4, 0, 0, 0, y3, y1, y2, 0, y3, y4 ]

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, y5, y3, y4, 0, 0, y6, y5, 0]

  p = - s 4 + s 7


 
1499 . Coloring, {2, 3, 4, 5, 6, 10, 11}

Ωp(Δ)=0:     p = 9s 3 - 16s 5 - 24s 6 - 16s 7 + 64s 8   p' = 3s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, C, B, 2, 4, 5]
B: [6, 7, 7, 6, A, A, A, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -7 y1 + 9 y2 - 7 y3 + 9 y4 - 7 y5 + 9 y6 - 7 y7, 7 y1, 7 y2, 7 y3, 0, 7 y4, 7 y5, 0, 0, 7 y6, 7 y7]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y5 - y1 - y4 + 3 y2 + y3, 0, 0, 0, 0, y5, 2 y2, 0, y1, y4, y2, y3]

  p = - s 5 + s 6   p = - s 5 + s 7


 
1500 . Coloring, {2, 3, 4, 5, 6, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8   p = 9s 3 + 8s 5 - 8s 6 + 16s 7 - 64s 8

R: [7, 8, 8, 7, 3, 3, B, C, B, 2, 1, 9]
B: [6, 7, 7, 6, A, A, A, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y2, y1, y1, 0, 0, 0, y7, y6, y5, 0, y4, y3]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, 2 y5, 0, 0, y4, y5, y6]

  p = s 4 - s 7


 
1501 . Coloring, {2, 3, 4, 5, 6, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, B, C, B, C, 4, 9]
B: [6, 7, 7, 6, A, A, A, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, 0, y4, y5, y3, 0, y2, y1]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, 0, 0, y2, y5, y4, 0, 0, y3, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1502 . Coloring, {2, 3, 4, 5, 7, 8, 9}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, A, A, B, C, C, 1, 5]
B: [6, 7, 7, 6, A, 3, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y8, 0, y7, 0, y6, 0, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y7, y6, y5, 0, y4, y3, 0, y7, y2, y1, y2]

  p' = s 3 - s 8   p = s 3 - s 8


 
1503 . Coloring, {2, 3, 4, 5, 7, 8, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, y8, 0, y5, 0, y4, y3, 0, y2, y1, 0]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 7 y7, 7 y6, 0, 7 y5, 7 y4, 0, 7 y3, 7 y2, -7 y7 - 7 y6 - 7 y5 - 7 y4 + 9 y3 + 9 y2 + 9 y1, 7 y1]

  p = - s 2 - s 3 + s 7 + s 8


 
1504 . Coloring, {2, 3, 4, 5, 7, 8, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, C, 4, 5]
B: [6, 7, 7, 6, A, 3, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y5, 0, y3, y4, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[y3, y1, y2, 0, 0, -y3 - y1 - y2 - y5 - y4 - y6 + 6 y7, y5, 0, y7, y4, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
1505 . Coloring, {2, 3, 4, 5, 7, 8, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, C, 1, 9]
B: [6, 7, 7, 6, A, 3, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y8, 0, y7, 0, 0, 0, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y9, y8, y7, y6, y5, y4, 0, 0, y3, y2, y1]


 
1506 . Coloring, {2, 3, 4, 5, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y8, y7, y6, 0, y5, 0, y4, y3, 0, y2, 0, y1]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y8, y7, 0, y6, y5, 0, y4, y3, y2, y1]


 
1507 . Coloring, {2, 3, 4, 5, 7, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, C, 4, 5]
B: [6, 7, 7, 6, A, 3, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y2, y3, y4, 0, y1, y6, 0, y7, 0, y5]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, 0, 2 y6, y6, y7, 0]

  p = - s 3 + s 8


 
1508 . Coloring, {2, 3, 4, 5, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, C, 1, 9]
B: [6, 7, 7, 6, A, 3, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, y1, 0, 0, 0, -2 y1 + 2 y5, y5, y4, y3, 0, y2]

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y4, y5, y3, y1, y2, 0, 0, y7, y8, 0]


 
1509 . Coloring, {2, 3, 4, 5, 7, 10, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, A, A, C, B, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y8, y6, y7, y5, 0, y4, y3, 0, y2, y1, y9]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[3 y3, 0, 3 y2, 0, 0, 3 y1, -3 y3 - 3 y2 - 3 y1 + 10 y6 + 10 y5 - 3 y4, 0, 3 y6, 3 y5, 3 y4, 3 y6 + 3 y5]

  p = - s 2 + s 7   p' = s 2 - s 7


 
1510 . Coloring, {2, 3, 4, 5, 7, 10, 12}

Ωp(Δ)=0:     p' = s 3 + 8s 5 - 8s 6 + 16s 7   p' = s 2 + 8s 4 - 8s 5 + 16s 6   p = s 2 - 8s 5 - 48s 6 + 64s 7 - 128s 8

R: [7, 8, 8, 7, 3, A, A, C, B, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, 0, y4, y9, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -3 y1 + 5 y2 - 3 y3 - 3 y4 + 5 y5 - 3 y6 + 5 y7, 3 y1, 3 y2, 3 y3, 3 y4, 0, 0, 3 y5, 3 y6, 3 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1511 . Coloring, {2, 3, 4, 5, 7, 11, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, A, C, B, C, 4, 9]
B: [6, 7, 7, 6, A, 3, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y8, y7, 0, 0, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y3, y2, y1, 0, y9, y8, y7, 0, 0, y6, y5, y4]


 
1512 . Coloring, {2, 3, 4, 5, 8, 9, 10}

Ωp(Δ)=0:     p = s 2 + 8s 5 - 48s 6 - 64s 7 - 128s 8   p' = s 2 + 8s 4 + 8s 5 + 16s 6   p' = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, A, B, B, C, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y6, y7, 0, y6, 0, y5, y4, 0, y3, y2, y3]

  p' = s 5 - s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, 0, y3, y2, y1, y7 + y6 - y5 - y4 + y3 + y2 - y1]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1513 . Coloring, {2, 3, 4, 5, 8, 9, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, 3, A, B, B, C, C, 4, 5]
B: [6, 7, 7, 6, A, 3, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y4, y5, 0, y2, y3, 0, y7, y8, y6]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, 0, y6, y7, y8, y9]


 
1514 . Coloring, {2, 3, 4, 5, 8, 9, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, B, B, C, C, 1, 9]
B: [6, 7, 7, 6, A, 3, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y5, 0, 3 y4, 0, 0, 0, 3 y3, 3 y2, 3 y1, 3 y4, -3 y5 + 7 y4 - 3 y3 - 3 y2 + 10 y1, 3 y4 + 3 y1]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y1, y7, y7, y6, y5, 0, 0, y4, y3, y3]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
1515 . Coloring, {2, 3, 4, 5, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 - 16s 5 - 8s 6 + 16s 7 + 64s 8   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, A, B, B, B, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 2 y1 - 3 y6, y2, 2 y6, 0, y4, y5, 0, y6, y3, 0]

  p = s 4 - s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[-y2 + y1 + y4 - y5 - y3 + y6, 0, y2, 0, 0, y1, y4, 0, y5, y3, 0, y6]

  p = - s 6 + s 7


 
1516 . Coloring, {2, 3, 4, 5, 8, 10, 12}

Ωp(Δ)=0:     p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7   p = s 3 - 8s 5 - 24s 6 - 16s 7 - 64s 8

R: [7, 8, 8, 7, 3, A, B, B, B, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y5, 0, 0, 0, y4, y3, 2 y5, y5, y6, 0]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y3, y4, y5, y1, y2, 0, 0, y6, 0, y7]


 
1517 . Coloring, {2, 3, 4, 5, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, B, B, C, 4, 9]
B: [6, 7, 7, 6, A, 3, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y6, y5, 0, 0, y4, y3, y2, y6, y1, y3]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y1, y2, 0, y3, y3, y4, 0, 0, y5, 0, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1518 . Coloring, {2, 3, 4, 5, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 8s 6 - 16s 7   p' = s 3 - 8s 5 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, A, B, C, C, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 9 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -2 y1 + 5 y4 - 2 y5, 2 y1, 2 y4, 2 y2, 0, 2 y4, 2 y3, 0, 5 y4 - 2 y2 - 2 y3, 2 y4, 2 y5]

  p = - s 3 + s 4 - s 5 + s 6   p = - s 3 + s 7   p' = - s 4 + s 8   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, -y1 + y7 + y6 - y5 - y4 + y3 + y2, 0, 0, y7, y6, 0, y5, y4, y3, y2]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1519 . Coloring, {2, 3, 4, 5, 9, 10, 12}

Ωp(Δ)=0:     p' = s 3 + 8s 6 + 16s 7   p = s 3 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, A, B, C, C, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y3, 5 y3 - 2 y2 - 2 y4, 2 y2, 0, 0, 0, 2 y3, 5 y3 - 2 y2 - 2 y1, 2 y1, 2 y2, 2 y3, 2 y4]

  p' = - s 6 + s 7   p = s 4 - s 7   p' = s 4 - s 6   p' = s 5 - s 6   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y5, y4, y3, y2, y1, 0, 0, y7, y6, y8]


 
1520 . Coloring, {2, 3, 4, 5, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, B, C, C, C, 4, 9]
B: [6, 7, 7, 6, A, 3, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 5 y2 - 2 y3, 2 y2, 0, 0, 2 y2, -2 y1 + 2 y3, 2 y1, 5 y2 - 2 y3, 2 y2, 2 y3]

  p = - s 3 + s 5   p = - s 3 + s 6   p = - s 3 + s 7   p = - s 3 + s 8   p = - s 3 + s 4

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y7, y4, y5, 0, 0, y6, y7, 0]

  p = - s 5 + s 8


 
1521 . Coloring, {2, 3, 4, 5, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 + 8s 5 + 32s 7   p = - 27s 2 + 56s 5 + 16s 6 + 256s 7 + 128s 8   p = 9s 2 + 2s 4 - 16s 5 - 80s 7 - 32s 8

R: [7, 8, 8, 7, 3, A, B, C, B, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y5, y1, 0, 0, y8, y7, y6, y5, y4, y3]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y3, 0, y2, y4, y5, 0, 0, y6, y7, y8]


 
1522 . Coloring, {2, 3, 4, 6, 7, 8, 9}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7   p = 3s 2 + 5s 3 + 10s 4 - 8s 6 - 32s 7 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, C, C, 1, 5]
B: [6, 7, 7, 6, 3, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y5, 0, y4, 0, y3, 0, y2, y1, 0, y8, y7, y6]

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y7, y8, y6, 0, y5, y4, 0, y3, y2, y1, y8]

  p = - s 3 + s 9


 
1523 . Coloring, {2, 3, 4, 6, 7, 8, 10}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 2 y3, 0, y5, y4, 0, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, 0, y5, y6, y7, y8]


 
1524 . Coloring, {2, 3, 4, 6, 7, 8, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, C, 4, 5]
B: [6, 7, 7, 6, 3, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-y1 - y2 - y3 - y4 + 6 y7 - y5 - y6, y1, y2, 0, 0, y3, y4, 0, y7, y5, y6, y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
1525 . Coloring, {2, 3, 4, 6, 7, 8, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, C, 1, 9]
B: [6, 7, 7, 6, 3, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, 0, y6, y5, y4, y3, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y7, y4, y5, y6, y3, y2, 0, 0, y1, y9, y8]


 
1526 . Coloring, {2, 3, 4, 6, 7, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y7, y6, y7, 0, y5, 0, y3, y4, 0, y2, 0, y1]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, 0, y5, y6, y7, y8]


 
1527 . Coloring, {2, 3, 4, 6, 7, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, C, 4, 5]
B: [6, 7, 7, 6, 3, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y2, 2 y2, y1, 0, -2 y2 + 2 y5, y5, 0, y4, 0, y3]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y5, y4, 0, 2 y3, y6, y7, 0]

  p = - s 2 + s 8


 
1528 . Coloring, {2, 3, 4, 6, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, C, 1, 9]
B: [6, 7, 7, 6, 3, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y1, 0, y1, 0, 0, 0, -2 y1 + 2 y4, y4, y5, y3, 0, y2]

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y5, y6, y7, y8, y4, y1, 0, 0, y2, y3, 0]


 
1529 . Coloring, {2, 3, 4, 6, 7, 10, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, A, C, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y8, y6, y7, y5, 0, y4, y3, 0, y2, y6, y1]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y2, 0, y1, 0, 0, y5, y4, 0, y3, y8, y7, y6]


 
1530 . Coloring, {2, 3, 4, 6, 7, 10, 12}

Ωp(Δ)=0:     p = s 2 - 8s 5 + 48s 6 + 128s 8   p' = s 2 - 8s 4 + 24s 5 - 16s 6 + 64s 7   p' = s 3 - 4s 4 + 8s 5 - 8s 6 + 16s 7

R: [7, 8, 8, 7, A, 3, A, C, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, 0, y4, y5, y8, y6, y7, y9]

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, y5, y4, y3, 0, 0, y2, y1, y8]


 
1531 . Coloring, {2, 3, 4, 6, 7, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, A, C, B, C, 4, 9]
B: [6, 7, 7, 6, 3, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y8, 0, 0, y3, y4, y5, y6, y2, y1]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y4, y1, y2, 0, y3, y5, y6, 0, 0, y7, y8, y9]


 
1532 . Coloring, {2, 3, 4, 6, 8, 9, 10}

Ωp(Δ)=0:     p = s 2 + s 3 + 24s 6 - 16s 7 + 64s 8   p' = - s 2 + 8s 4 + 24s 5 + 16s 6 + 64s 7   p' = s 2 + 2s 3 - 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, B, B, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y5, y6, y8, 0, y1, 0, y2, y3, 0, y4, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, 0, y3, y1, y2, y7 + y6 - y5 - y4 + y3 + y1 - y2]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1533 . Coloring, {2, 3, 4, 6, 8, 9, 11}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, B, B, C, C, 4, 5]
B: [6, 7, 7, 6, 3, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y3, y2, y1, 0, 5 y1 - y4 + 4 y3 + 4 y2 - 6 y5, y4, 0, -2 y3 - 2 y2 - 4 y1 + 5 y5, y5, 6 y1 + 5 y3 + 5 y2 - 8 y5]

  p = s 3 - s 6   p' = s 3 - s 6   p' = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y8, 0, 0, y3, y4, 0, y5, y6, y7, y8]

  p = - s 6 + s 9


 
1534 . Coloring, {2, 3, 4, 6, 8, 9, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, B, B, C, C, 1, 9]
B: [6, 7, 7, 6, 3, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 9 8 vs 9 5 vs 8 6 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, 0, 3 y3, 0, 0, 0, -3 y1 - 3 y5 + 7 y3 - 3 y2 + 10 y4, 3 y5, 3 y4, 3 y3, 3 y2, 3 y4 + 3 y3]

  p' = s 4 - s 7   p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, y5, y4, y4, y3, y2, 0, 0, y1, -y5 + y3, -y5 + y3]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = s 4 - s 7

 ยป SYNC'D !RANK'D


 
1535 . Coloring, {2, 3, 4, 6, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 - 8s 6 - 16s 7   p' = s 3 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, 3, B, B, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 4 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y3, y2, y1, 2 y2, 0, y7, y6, 0, y5, y4, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, y4, 0, 0, -2 y4 + 2 y3, y3, 0, y2, y1, 0, 5 y4 - 3 y3 + y2 + y1]

  p' = - s 4 + s 6   p' = - s 4 + s 5   p = s 4 - s 5


 
1536 . Coloring, {2, 3, 4, 6, 8, 10, 12}

Ωp(Δ)=0:     p' = s 3 + 8s 5 + 8s 6 + 16s 7   p = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, 3, B, B, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, y2, y5, 0, 0, 0, y3, y4, 2 y5, y5, y6, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y6, y7, y5, y4, 0, 0, y3, 0, y2]


 
1537 . Coloring, {2, 3, 4, 6, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, B, B, B, C, 4, 9]
B: [6, 7, 7, 6, 3, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y2, y6, 0, 0, y5, y4, y3, y2, y1, y4]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y4, y5, y6, 0, y2, y2, y3, 0, 0, y1, 0, y4]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1538 . Coloring, {2, 3, 4, 6, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + s 4 + 4s 5 + 8s 7 - 16s 8   p = - 9s 3 - 8s 5 + 8s 6 - 16s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, B, C, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, y1, -y1 + 5 y6 - y2 - y3 - y4 - y5, y6, y2, 0, y6, y3, 0, y4, y6, y5]

  p' = s 2 - s 7   p' = s 3 - s 8   p = s 2 - s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, y4, 0, 0, y5, y6, 0, -y7 - y4 + y5 + y6 - y1 + y2 + y3, y1, y2, y3]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1539 . Coloring, {2, 3, 4, 6, 9, 10, 12}

Ωp(Δ)=0:     p = 9s 3 - 16s 5 - 24s 6 - 16s 7 + 64s 8   p' = 3s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, 3, B, C, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y2, 2 y3, 2 y4, 0, 0, 0, 2 y2, 5 y2 - 2 y4 - 2 y1, 2 y1, 2 y4, 2 y2, 5 y2 - 2 y3 - 2 y4]

  p' = - s 5 + s 7   p' = - s 5 + s 8   p' = s 4 - s 5   p' = - s 5 + s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y1, y3, y7, y8, 0, 0, y6, y5, y4]


 
1540 . Coloring, {2, 3, 4, 6, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, B, C, C, C, 4, 9]
B: [6, 7, 7, 6, 3, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 3 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 2 y3, 2 y2, 0, 0, 2 y2, -2 y1 - 2 y3 + 5 y2, 2 y1, 2 y3, 2 y2, -2 y3 + 5 y2]

  p' = - s 6 + s 7   p' = s 3 - s 6   p' = s 5 - s 6   p' = s 4 - s 6   p = s 3 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y3, y1, 0, 2 y2 - 2 y1, y5, y6, 0, 0, y4, 2 y2 - 2 y1, 0]

  p = s 4 - s 7   p' = s 4 - s 7


 
1541 . Coloring, {2, 3, 4, 6, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 8s 4 - 8s 5 - 16s 6   p' = s 3 - 8s 5 - 8s 6 - 16s 7   p = s 2 - 8s 5 - 80s 6 - 64s 7 - 128s 8

R: [7, 8, 8, 7, A, 3, B, C, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y6, y5, 0, 0, y4, y3, y2, y6, y8, y7]

  p = s 6 - s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y3, 0, y2, 0, y1, y8, y7, 0, 0, y5, y6, y4]


 
1542 . Coloring, {2, 3, 4, 7, 8, 9, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y5, y4, 0, 0, y3, 0, y2, y1, 0, y8, y7, y6]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y7, y6, 0, y5, y4, 0, y3, 0, y2, y1]


 
1543 . Coloring, {2, 3, 4, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, C, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, 0, y3, y7, 0, y4, y5, y6]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 2 y7, y2, 0, 0, y3, y4, 0, y5, 0, y6, y7]

  p = s 3 - s 8


 
1544 . Coloring, {2, 3, 4, 7, 8, 9, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, C, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, 0, y2, y3, y4, y5, y6, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y7, y5, y4, y1, y2, y3, 0, 0, 0, y6, y7]

  p = - s 3 + s 8


 
1545 . Coloring, {2, 3, 4, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, A, A, B, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 2 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y6, 0, y5, y4, 0, y3, y2, 0, y1, -y6 + y5 - y4 - y3 + y2 + y1, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[y2, 0, y2, 0, 0, y2, y2, 0, 3 y2 - y1, 0, y2, y1]

  p' = - s 2 + s 6   p = s - s 3   p' = s - s 3   p' = - s 2 + s 4   p' = - s 3 + s 5


 
1546 . Coloring, {2, 3, 4, 7, 8, 10, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, B, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y6, y5, 0, 0, 0, 0, y4, y3, y2, y1, y7, 0]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y7, y6, y5, y4, y3, 0, 0, 0, y2, y1]


 
1547 . Coloring, {2, 3, 4, 7, 8, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, B, B, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, 0, y5, y6, y2, y3, y4, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, y6, y4, 0, y1, y2, y3, 0, 0, 0, y5, y6]

  p = - s 3 + s 8


 
1548 . Coloring, {2, 3, 4, 7, 9, 10, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, C, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y7, 0, y6, y5, 0, y4, y3, 0, y2, 0, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y3, 0, y2, 0, 0, y1, y7, 0, y6, 0, y5, y4]


 
1549 . Coloring, {2, 3, 4, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, A, A, C, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 0, 0, y7, y3, y6, y5, 0, y4]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y5, y4, y3, y2, 0, 0, 0, y7, y6]


 
1550 . Coloring, {2, 3, 4, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, A, A, C, C, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 6 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 0, y5, 0, 0, y4, y5, y3, y2, 0, y1]

  p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y3, y4, 0, y3, y2, y6, 0, 0, 0, y5, 0]

  p = - s 2 + s 7


 
1551 . Coloring, {2, 3, 4, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, C, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y7, 0, y8, 0, 0, y1, y2, y3, y4, y5, y6]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y4, 0, y5, 0, y6, y7, y1, 0, 0, 0, y2, y3]


 
1552 . Coloring, {2, 3, 4, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 24s 6 - 16s 7 - 64s 8   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, A, B, B, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y7, 0, y5, y6, 0, y4, y3, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y3 - y7 - y4 + y5 + y6, 0, y1, 0, 0, y2, y3, 0, y7, y4, y5, y6]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1553 . Coloring, {2, 3, 4, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 16s 5 - 8s 6 + 16s 7 + 64s 8   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, A, B, B, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 3 y1, 0, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, 0, 0, y3, y2, y1]


 
1554 . Coloring, {2, 3, 4, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, B, B, C, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, 2 y4, 0, 0, 2 y5, 7 y4 + 7 y5 - 9 y1 + 7 y2 - 9 y3, 2 y1, 7 y4 + 7 y5 - 9 y1 + 7 y2 - 9 y3, 2 y2, 2 y3]

  p = - s 2 - s 3 + s 5 + s 6   p' = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y4, y5, 0, y3, y1, y2, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1555 . Coloring, {2, 3, 4, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 + 8s 5 - 48s 6 - 64s 7 - 128s 8   p' = s 2 + 8s 4 + 8s 5 + 16s 6   p' = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, A, B, B, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y3, 0, y2, 0, 0, y1, y6, y5, y5, y4, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y7, 0, y5, y6, y4, 0, 0, y3, 0, y2]


 
1556 . Coloring, {2, 3, 4, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 9s 2 + 2s 4 - 16s 6 + 16s 7 - 96s 8   p = - 6s 2 + s 3 + 8s 6 - 16s 7 + 64s 8   p = - 27s 2 + 8s 5 + 48s 6 - 64s 7 + 256s 8

R: [7, 8, 8, 7, A, A, B, C, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, 0, y2, -y1 - y3 - y4 + 5 y2 - y5, y3, y4, y2, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6   p = - s 4 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y5, 0, y4, 0, y6, y7, y8, 0, 0, y2, y1, y3]


 
1557 . Coloring, {2, 3, 5, 6, 7, 8, 9}

R: [7, 8, 8, 6, 3, 3, A, B, C, C, 1, 5]
B: [6, 7, 7, 7, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y7, 0, y2, y3, y4, y5, 0, y6, y9, y8]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, 0, y3, 0, y6, y1, 0, y4, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1558 . Coloring, {2, 3, 5, 6, 7, 8, 10}

Ωp(Δ)=0:     p = 6s 2 + s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, 2, 1, 5]
B: [6, 7, 7, 7, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y8, y6, y7, 0, 2 y5, y5, y4, y3, 0, y2, y1, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 0, 7 y2, 0, 9 y2 + 9 y1 - 7 y3 - 7 y4 + 9 y5 - 7 y6, 7 y1, 0, 7 y3, 7 y4, 7 y5, 7 y6]

  p = s 3 + s 4 - s 6 - s 7


 
1559 . Coloring, {2, 3, 5, 6, 7, 8, 11}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, C, 4, 5]
B: [6, 7, 7, 7, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 9 vs 9 4 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[y1, -y1 + 2 y3, 0, 0, 0, -y4 + 2 y3, y4, 0, y3, y2, 2 y3 - y2, y3]

  p' = - s + s 7   p' = - s 2 + s 3 - s 5 + s 6   p' = s - s 2 + s 4 - s 5   p = s - s 3 + s 4 - s 6


 
1560 . Coloring, {2, 3, 5, 6, 7, 8, 12}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, C, 1, 9]
B: [6, 7, 7, 7, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y3, 0, y2, 0, 0, y1, y9, y8, y7, y6, y5, y4]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y7, 0, y5, y6, y4, y3, 0, 0, y1, y2, 2 y4]

  p = s 5 - s 8


 
1561 . Coloring, {2, 3, 5, 6, 7, 9, 10}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, 2, 1, 5]
B: [6, 7, 7, 7, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 8 vs 9 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[2 y4, y2, y1, 0, y3, y4, y6, y5, 0, y7, 0, y8]

  p = - s 5 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, y6, y5, 0, y4, y3, y2, y1]


 
1562 . Coloring, {2, 3, 5, 6, 7, 9, 11}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, C, 4, 5]
B: [6, 7, 7, 7, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y2, 2 y3, y1, -3 y3 + 2 y5, y3, y4, 0, y5, 0, y6]

  p = s 3 - s 7   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y4, y3, 0, 0, 0, y2, y1, 0, y7, y6, y5, 0]


 
1563 . Coloring, {2, 3, 5, 6, 7, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, C, 1, 9]
B: [6, 7, 7, 7, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[-6 y3 + 8 y4 - 4 y1, 0, -4 y3 + 6 y4 - 3 y1, 0, 0, -3 y3 + 4 y4 - 2 y1, y3, y4, y5, y1, 0, y2]

  p = s 4 - s 6   p' = - s 5 + s 7   p' = s 4 - s 6

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, 0, y5, 2 y4, y4, y3, 0, 0, y2, y1, 0]

  p = s 4 - s 7


 
1564 . Coloring, {2, 3, 5, 6, 7, 10, 11}

R: [7, 8, 8, 6, 3, 3, A, C, B, 2, 4, 5]
B: [6, 7, 7, 7, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 10 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, y2 + y6, y1 - y6 + y3 - y5, y4, y1, y2, y6, y3, 0, y4, y6, y5]

  p = s 4 - s 5 + s 6 - s 7   p = - s 4 + s 8   p' = - s 4 + s 8   p = s 4 - s 5 - s 7 + s 10

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y6, y5, 0, y4, y3, y2, y1]


 
1565 . Coloring, {2, 3, 5, 6, 7, 10, 12}

R: [7, 8, 8, 6, 3, 3, A, C, B, 2, 1, 9]
B: [6, 7, 7, 7, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 4 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, y9, y10]

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y2, -y1 + y3, y1, y4, 0, 0, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = s 3 - s 6


 
1566 . Coloring, {2, 3, 5, 6, 7, 11, 12}

R: [7, 8, 8, 6, 3, 3, A, C, B, C, 4, 9]
B: [6, 7, 7, 7, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y9, y8, 0, y7, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y1 - y7 - y6 + y5 + y4 - y3 + y2, 0, 0, y7, y6, y5, 0, 0, y4, y3, y2]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1567 . Coloring, {2, 3, 5, 6, 8, 9, 10}

R: [7, 8, 8, 6, 3, 3, B, B, C, 2, 1, 5]
B: [6, 7, 7, 7, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, 2 y4, y6, 0, y5, y4, y3, y2, 0, 0, y1, y4]

  p' = - s 5 + s 8   p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 + y2 - y6 - y3 + y4 + y5, 0, y1, y2, 0, y6, y3, y4, y5]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1568 . Coloring, {2, 3, 5, 6, 8, 9, 11}

R: [7, 8, 8, 6, 3, 3, B, B, C, C, 4, 5]
B: [6, 7, 7, 7, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, 0, y7, 3 y5]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y8, y7, 0, y3, y4, y5, y6]


 
1569 . Coloring, {2, 3, 5, 6, 8, 9, 12}

R: [7, 8, 8, 6, 3, 3, B, B, C, C, 1, 9]
B: [6, 7, 7, 7, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-5 y1 - 5 y2 - 5 y7 - 5 y3 + 11 y4 - 5 y5 + 11 y6, 0, 5 y1, 0, 0, 5 y2, 5 y7, 5 y3, 5 y4, 0, 5 y5, 5 y6]

  p = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, 0, y5, y5, y4, y3, 0, 0, y2, y4, y4]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = s 4 - s 7


 
1570 . Coloring, {2, 3, 5, 6, 8, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, 2, 4, 5]
B: [6, 7, 7, 7, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 4 vs 6

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y3, y1, y2, 2 y3, y4, y3, y5, 0, 0, y6, 0]

  p = s 2 - s 7   p' = s 2 - s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[-2 y1 + 2 y2 + 2 y3 - 2 y4, 0, 0, 0, 0, y1, -3 y1 + 3 y2 + 3 y3 - 3 y4, 0, y2, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6

 ยป SYNC'D !RANK'D


 
1571 . Coloring, {2, 3, 5, 6, 8, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, 2, 1, 9]
B: [6, 7, 7, 7, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 5 vs 6

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y3, y2, 0, 0, y3, y4, y5, 2 y3, 0, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y2, y3, y2, y1, 0, 0, y5, 0, y4]

  p = s 3 - s 6

 ยป SYNC'D !RANK'D


 
1572 . Coloring, {2, 3, 5, 6, 8, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, C, 4, 9]
B: [6, 7, 7, 7, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y3, y2, 0, y1, y4, y6, y5, 0, y7, 2 y4]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[2 y2 - 2 y5, y1, 0, 0, y2, y5, y4, 0, 0, y3, 0, 2 y2 - 2 y5]

  p' = - s 3 + s 6   p = s 3 - s 6


 
1573 . Coloring, {2, 3, 5, 6, 9, 10, 11}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, 2, 4, 5]
B: [6, 7, 7, 7, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 2 y3, y6, y5, y1, y2, y3, y4, 0, 0, y7, y8]

  p = - s 5 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, 0, y1, y2 - y1 + y6 + y5 - y4 - y3, 0, y6, y5, y4, y3]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1574 . Coloring, {2, 3, 5, 6, 9, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, 2, 1, 9]
B: [6, 7, 7, 7, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y1, 10 y2, 11 y1 - 15 y2 + 11 y3 - 5 y4 - 5 y5 + 11 y6 - 5 y7, 0, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7]

  p' = - s 4 - s 5 + s 7 + s 8   p = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, y6, y4, y3, 0, 0, y2, 2 y4, y1]

  p = s 4 - s 7


 
1575 . Coloring, {2, 3, 5, 6, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, C, 4, 9]
B: [6, 7, 7, 7, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 8 vs 8 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y8, 0, y7, y6, y5, y4, 0, y2, y3]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y6, y4, y3, 0, 0, y5, y6, 0]

  p = - s 4 + s 7


 
1576 . Coloring, {2, 3, 5, 6, 10, 11, 12}

R: [7, 8, 8, 6, 3, 3, B, C, B, 2, 4, 9]
B: [6, 7, 7, 7, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 2 y5, y8, y6, 0, y7, y5, y4, y3, 0, y1, y2]

  p = - s 2 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 3 y5, 0, 0, y4, y5, y6]

  p = - s 4 + s 7


 
1577 . Coloring, {2, 3, 5, 7, 8, 9, 10}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, 2, 1, 5]
B: [6, 7, 7, 7, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 9 vs 10 6 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y3, y2, y1, 0, y9, y8, y7, y6, 0, y5, y4, y8]

  p = s 4 - s 10

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -2 y4 + y2, y1, 0, y4, y6, 0, y5, y4, y3, y2]

  p = s 4 - s 7   p' = s 4 - s 7

 ยป SYNC'D !RANK'D


 
1578 . Coloring, {2, 3, 5, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, C, 4, 5]
B: [6, 7, 7, 7, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 9 vs 9 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y5, y4, y1, y2, y3, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y6, y7, 0, 0, y4, y5, 0, y6, y3, y2, y3]

  p' = - s 3 + s 8   p = - s 3 + s 8


 
1579 . Coloring, {2, 3, 5, 7, 8, 9, 12}

Ωp(Δ)=0:     p = 2s 2 + 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, C, 1, 9]
B: [6, 7, 7, 7, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y5, 0, 0, y5, y4, y2, y3, y8, y6, y7]

  p = s 7 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y3 - y7, y2, y3, y7, y4, 0, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8


 
1580 . Coloring, {2, 3, 5, 7, 8, 10, 11}

R: [7, 8, 8, 6, 3, A, A, B, B, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y4, 2 y5, y3, y5, y7, 0, y6, y8, 0]

  p = s 3 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[-7 y1 - 7 y2 - 7 y3 + 9 y4 + 9 y5 - 7 y6 + 9 y7, 0, 7 y1, 0, 0, 7 y2, 7 y3, 0, 7 y4, 7 y5, 7 y6, 7 y7]

  p = - s 2 - s 3 + s 7 + s 8


 
1581 . Coloring, {2, 3, 5, 7, 8, 10, 12}

R: [7, 8, 8, 6, 3, A, A, B, B, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y3, y5, y4, 2 y3, y7, y6, 0]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 3 y5, 3 y4, 3 y3, 3 y2, 3 y5 + 3 y4, 0, 0, 3 y1, 3 y5 + 3 y4 - 3 y2, 7 y5 + 7 y4 - 3 y3 - 3 y1]

  p = s 3 - s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
1582 . Coloring, {2, 3, 5, 7, 8, 11, 12}

R: [7, 8, 8, 6, 3, A, A, B, B, C, 4, 9]
B: [6, 7, 7, 7, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y2, y3, y4, y5, y8, y6, y7]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y3, y2, y1, 0, y9, y8, y7, 0, 0, y6, y5, y4]


 
1583 . Coloring, {2, 3, 5, 7, 9, 10, 11}

R: [7, 8, 8, 6, 3, A, A, C, C, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y1, y2, 2 y4, y6, y3, y4, y5, 0, y8, 0, y7]

  p = - s 5 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, 0, y4, y3, 0, y8, y7, y6, y5]


 
1584 . Coloring, {2, 3, 5, 7, 9, 10, 12}

R: [7, 8, 8, 6, 3, A, A, C, C, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y6, y7, y6, 0, 0, y6, y5, y4, y3, y2, 0, y1]

  p' = s 6 - s 8   p = s 6 - s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, -3 y5 - 7 y3 + 14 y2 + 8 y1, 3 y5, -5 y3 + 7 y2 + 7 y1, 3 y4, 3 y3, 0, 0, 3 y2, -3 y4 - 7 y3 + 8 y2 + 14 y1, 3 y1]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7


 
1585 . Coloring, {2, 3, 5, 7, 9, 11, 12}

R: [7, 8, 8, 6, 3, A, A, C, C, C, 4, 9]
B: [6, 7, 7, 7, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, -3 y5 + 2 y4, y5, y4, y3, y2, 0, y1]

  p' = s 5 - s 7   p' = s 4 - s 6   p = s 4 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, 0, 0, y7, y8, 0]


 
1586 . Coloring, {2, 3, 5, 7, 10, 11, 12}

Ωp(Δ)=0:     p = - 2s 2 - s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, C, B, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y9, y7, y8, 0, y6, y7, y5, y3, y4, y2, y1]

  p = - s 2 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[3 y2, 0, -3 y2 + 5 y1 - 3 y7 - 3 y6 + 5 y5 - 3 y3 + 5 y4, 0, 3 y1, 3 y7, 3 y6, 0, 0, 3 y5, 3 y3, 3 y4]

  p = s + s 2 + s 3 - s 6 - s 7 - s 8


 
1587 . Coloring, {2, 3, 5, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y7, y8, y6, y5, y3, y4, y2, 0, y1, -y7 - y8 + y6 + y5 - y3 + y2 + y1, y4]

  p = s 4 - s 10   p' = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[-y3 + y1 + y2 - y4 - y5 + y6 + y7, 0, y3, 0, 0, y1, y2, 0, y4, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1588 . Coloring, {2, 3, 5, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 10 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y8 - 3 y7 - 3 y6 + 13 y5 - 3 y4 - 3 y3 + 13 y2, 3 y8, 0, 0, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2]

  p' = s 5 + s 6 - s 8 - s 9   p = s 5 - s 7 - s 8 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y2, y2 + y6, y1, y6, y5, 0, 0, y4, y6, y3]

  p' = - s 4 + s 7   p = s 4 - s 7

 ยป SYNC'D !RANK'D


 
1589 . Coloring, {2, 3, 5, 8, 9, 11, 12}

Ωp(Δ)=0:     p = - 2s 2 + 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, C, 4, 9]
B: [6, 7, 7, 7, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 7 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y3, y2, 0, y1, y3, -y2 + y1 - y7 - y6 + y5 + y4, y7, y6, y5, y4]

  p' = s 7 - s 8   p = s 7 - s 9

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, y4, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

 ยป SYNC'D !RANK'D


 
1590 . Coloring, {2, 3, 5, 8, 10, 11, 12}

R: [7, 8, 8, 6, 3, A, B, B, B, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y6, y5, y4, 0, y3, y5, y2, 2 y5, y1, -y6 + 2 y5 + y4 - y3 + y2 + y1, 0]

  p' = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p' = s 2 - s 8   p = s 2 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y7, 0, y6, y5, y4, 0, 0, y3, 0, y2]


 
1591 . Coloring, {2, 3, 5, 9, 10, 11, 12}

R: [7, 8, 8, 6, 3, A, B, C, C, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1 - y2 + y4 + y5 + y6 - y7 - y8, y3, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 8 + s 9   p = - s 8 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]


 
1592 . Coloring, {2, 3, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = 9s 3 - 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, 2, 1, 5]
B: [6, 7, 7, 7, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y4 - y8, 0, y4, y8, y3, y7, 0, y5, y6, y8]

  p' = s 3 - s 9   p = s 3 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y7, y6, 0, y7, y5, 0, y4, y3, y2, y1]

  p = - s 5 + s 8


 
1593 . Coloring, {2, 3, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 9s 3 + 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, C, 4, 5]
B: [6, 7, 7, 7, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 7 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 0, -3 y6 - 3 y5 - 3 y2 - 3 y1 + 7 y4 + 7 y3, 3 y6, 3 y4, 3 y5, 3 y3, 3 y2, 0, 3 y4 + 3 y3, 3 y1, 3 y4 + 3 y3]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y1 - 2 y2 - y7 + y6 + y5 + y4 - y3, y2, 0, 0, y7, y6, 0, y5, y4, y3, y2]

  p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1594 . Coloring, {2, 3, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = 6s 2 + 5s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, C, 1, 9]
B: [6, 7, 7, 7, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y9, 0, y8, 0, 0, y7, y6, y4, y5, y3, y1, y2]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5, y6, y4, -y5 + y6 + 3 y3, y3, y2, 0, 0, -y5 + y6 + 2 y3, y1, y3]

  p' = - s 4 + s 7   p = - s 4 + s 7   p' = - s 5 + s 8


 
1595 . Coloring, {2, 3, 6, 7, 8, 10, 11}

R: [7, 8, 8, 6, A, 3, A, B, B, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y7, 2 y6, y8, y6, y5, 0, y4, y3, 0]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y1 - y5 + y4 + y3 + y2 - y7 - y6, 0, 0, y5, y4, 0, y3, y2, y7, y6]

  p = s 7 - s 8


 
1596 . Coloring, {2, 3, 6, 7, 8, 10, 12}

R: [7, 8, 8, 6, A, 3, A, B, B, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, 2 y4, y7, y8, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y3, y2, y1, y8, 0, 0, y7, y6, y5]


 
1597 . Coloring, {2, 3, 6, 7, 8, 11, 12}

R: [7, 8, 8, 6, A, 3, A, B, B, C, 4, 9]
B: [6, 7, 7, 7, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y4, 0, y2, y3, y7, y5, y6, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y2, y2 - y1 + y8 - y6 + y7 + y3 - y4 - y5, y1, 0, y8, y6, y7, 0, 0, y3, y4, y5]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1598 . Coloring, {2, 3, 6, 7, 9, 10, 11}

R: [7, 8, 8, 6, A, 3, A, C, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y4, y5, 2 y2, y6, y1, y2, y3, 0, y7, 0, y8]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y1 - y7 + y6 + y5 + y4 - y3 - y2, 0, 0, y7, y6, 0, y5, y4, y3, y2]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1599 . Coloring, {2, 3, 6, 7, 9, 10, 12}

R: [7, 8, 8, 6, A, 3, A, C, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[4 y2 - 2 y3, y1, y2, 0, 0, 2 y2 - y3, y3, y7, y4, y5, 0, y6]

  p = - s 6 + s 8   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y5, y3, y1, y2, 0, 0, y8, y7, y6]


 
1600 . Coloring, {2, 3, 6, 7, 9, 11, 12}

R: [7, 8, 8, 6, A, 3, A, C, C, C, 4, 9]
B: [6, 7, 7, 7, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, -8 y1 + 3 y4 + y2 + y3 - y5, 2 y1, 0, -5 y1 + 2 y4, y1, y2, y3, y4, 0, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y4, y2, y3, 0, -y4 + y2 + y3 + y5 - y1 - y6 + y7, y5, y1, 0, 0, y6, y7, 0]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1601 . Coloring, {2, 3, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, C, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y3, y1, y2, 0, y4, y5, y6, y7, y8, y9, y10]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y7, 0, y6, 0, y4, y5, y3, 0, 0, y2, y7 - y6 + y4 - y5 + y3 + y2 - y1, y1]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1602 . Coloring, {2, 3, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y9, y6, 0, y7, y8, y9]

  p = s 5 - s 10

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, y6, 0, 0, y5, y4, 0, y3, y2, y1, y7 + y6 - y5 - y4 + y3 + y2 - y1]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1603 . Coloring, {2, 3, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-6 y1 - 9 y5 - 3 y2 - 3 y3 + 13 y4 - 3 y6 + 13 y7, 3 y1 + 3 y5, 3 y1, 0, 0, 3 y5, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 + s 6 + s 7 - s 9   p' = - s 4 - s 5 + s 7 + s 8   p = - s 4 + s 10

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, y3, y2, y6, y1, 0, 0, y7, y6, y5]

  p = - s 3 + s 8


 
1604 . Coloring, {2, 3, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, C, 4, 9]
B: [6, 7, 7, 7, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -3 y1 - 3 y6 + 7 y4 - 3 y2 + 10 y3 - 3 y5, 3 y1, 0, 3 y6, 3 y4, 3 y2, 3 y3, 3 y4, 3 y5, 3 y4 + 3 y3]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, y2, y3, y4, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1605 . Coloring, {2, 3, 6, 8, 10, 11, 12}

R: [7, 8, 8, 6, A, 3, B, B, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, y6, y5, 2 y6, y6, y7, 0]

  p = - s 3 + s 8   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y4, 0, y3, 0, y2, y1, y6, 0, 0, y5, 0, y7]


 
1606 . Coloring, {2, 3, 6, 9, 10, 11, 12}

R: [7, 8, 8, 6, A, 3, B, C, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y4, y2, y1, 0, y2 + y1 - y6 - y5 + y3, y7, y6, y5, y7, y4, y3]

  p' = s 8 - s 9   p' = s 7 - s 9   p = s 7 - s 10

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]


 
1607 . Coloring, {2, 3, 7, 8, 9, 10, 11}

R: [7, 8, 8, 6, A, A, A, B, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y4, 0, y2, y3, y1, y7, y8, 0, y5, y6, y7]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, 0, y5, y4, 0, y3, 0, y1, y2]


 
1608 . Coloring, {2, 3, 7, 8, 9, 10, 12}

R: [7, 8, 8, 6, A, A, A, B, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[11 y1 + 11 y2 + 11 y3 - 2 y7 - 2 y4 + 11 y5 - 39 y6, 2 y1, 0, 0, 0, 2 y2, 2 y3, 2 y7, 3 y1 + 3 y2 + 3 y3 + 3 y5 - 11 y6, 2 y4, 2 y5, 2 y6]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y4, y5, y6, y3, y2, 0, 0, 0, y1, 3 y3]

  p = - s 4 + s 7


 
1609 . Coloring, {2, 3, 7, 8, 9, 11, 12}

R: [7, 8, 8, 6, A, A, A, B, C, C, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, y2, y1, 2 y1, y7, y6, y5, y4]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y5, 2 y7, y4, 0, y2, y3, y1, 0, 0, 0, y6, y7]

  p = - s 3 + s 8


 
1610 . Coloring, {2, 3, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, B, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, 0, y3, y4, y5, 2 y4, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, 0, y7, y6]


 
1611 . Coloring, {2, 3, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, A, A, C, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y2, 0, 2 y6, 0, y1, y6, y7, y5, y4, 0, y3]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y5, y3, y4, 0, 0, 0, y6, y7]


 
1612 . Coloring, {2, 3, 8, 9, 10, 11, 12}

R: [7, 8, 8, 6, A, A, B, B, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y7, 0, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y6 - 3 y4 - 3 y3 + 8 y2 + 5 y1, -3 y7 - 3 y5 + 5 y2 + 8 y1, 3 y1]

  p' = s 2 - s 8   p = s 2 - s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y5, y3, y4, 0, 0, y8, y7, y6]


 
1613 . Coloring, {2, 4, 5, 6, 7, 8, 9}

R: [7, 8, 7, 7, 3, 3, A, B, C, C, 1, 5]
B: [6, 7, 8, 6, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y8, 0, y7, 0, y6, y5, y4, y3, y2, y1, -y8 + y7 - y6 + y5 + y4 + y3 + y2 - y1]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1614 . Coloring, {2, 4, 5, 6, 7, 8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, 2, 1, 5]
B: [6, 7, 8, 6, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, y4, 0, y2, y3, 0, y8, y1, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y2, y3, y3, y4, y5, y6, y1 - y2 + 2 y3 + y4 + y5 - y6]

  p = - s 6 + s 8   p = - s 6 + s 7


 
1615 . Coloring, {2, 4, 5, 6, 7, 8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, C, 4, 5]
B: [6, 7, 8, 6, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y3, y1, 0, y6, y4, 0, y5, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[8 y1 + 8 y2 - 3 y3 - 11 y4 - 11 y5 - 3 y6 + 8 y7, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 5 y1 + 5 y2 - 8 y4 - 8 y5 + 5 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8


 
1616 . Coloring, {2, 4, 5, 6, 7, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, C, 1, 9]
B: [6, 7, 8, 6, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 2 y7, 0, 0, 0, y6, y7, y4, y3, y5, y2]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
1617 . Coloring, {2, 4, 5, 6, 7, 9, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, 2, 1, 5]
B: [6, 7, 8, 6, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y1, y2, y7, 0, y3, 0, y4, y5, 0, y6, 0, y8]

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - 2 y6 - y2 - y3 + y4 + y5, 0, y1, y6, y6, y2, y3, y4, y5]

  p' = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = s 2 - s 8


 
1618 . Coloring, {2, 4, 5, 6, 7, 9, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, C, 4, 5]
B: [6, 7, 8, 6, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y4, 2 y1, y3, 0, y2, y1, 0, y5, 0, y6]

  p = - s 2 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y4 - y5 - 3 y6 - y2 + y3, y1, 0, 0, 0, y4, y5, y6, 2 y6, y2, y3, 0]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8


 
1619 . Coloring, {2, 4, 5, 6, 7, 9, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, C, 1, 9]
B: [6, 7, 8, 6, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, 2 y4, 0, 0, 0, y5, y4, y3, y2, 0, y1]

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y4, 0, y3, 2 y7, y1, y2, y7, 0, y6, y5, 0]

  p = - s 2 + s 8


 
1620 . Coloring, {2, 4, 5, 6, 7, 10, 11}

R: [7, 8, 7, 7, 3, 3, A, C, B, 2, 4, 5]
B: [6, 7, 8, 6, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y9, y8, y7, y6, 0, y5, y3, 0, y4, y2, y1]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, 0, y5, y4, y4, y3, y2, y6 - y5 + 2 y4 + y3 + y2 - y1, y1]

  p' = - s 6 + s 7   p = s 6 - s 7


 
1621 . Coloring, {2, 4, 5, 6, 7, 10, 12}

R: [7, 8, 7, 7, 3, 3, A, C, B, 2, 1, 9]
B: [6, 7, 8, 6, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, y3, 0, 0, 0, y4, y5, y6, y7, y9, y8]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y5, y6, y7, y7, 0, y1, y2, y3]

  p = - s 5 + s 8


 
1622 . Coloring, {2, 4, 5, 6, 7, 11, 12}

R: [7, 8, 7, 7, 3, 3, A, C, B, C, 4, 9]
B: [6, 7, 8, 6, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 2 y3, y1, 0, 0, y2, y3, y4, y5, y6, y7]

  p = s 2 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y8, y7, 0, 0, y6, y5, y4, y3, 0, y2, y1, y3]

  p = s 3 - s 9


 
1623 . Coloring, {2, 4, 5, 6, 8, 9, 10}

R: [7, 8, 7, 7, 3, 3, B, B, C, 2, 1, 5]
B: [6, 7, 8, 6, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y6, y2, 0, y3, 0, y4, 2 y3 - 3 y6, 0, 0, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, y1, y6, y6, y5, y3, y4, y2]

  p = - s 2 + s 8


 
1624 . Coloring, {2, 4, 5, 6, 8, 9, 11}

R: [7, 8, 7, 7, 3, 3, B, B, C, C, 4, 5]
B: [6, 7, 8, 6, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, 0, y6, 3 y5]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8, y9]


 
1625 . Coloring, {2, 4, 5, 6, 8, 9, 12}

R: [7, 8, 7, 7, 3, 3, B, B, C, C, 1, 9]
B: [6, 7, 8, 6, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 6 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-5 y1 - 15 y2 + 11 y3 - 5 y5 + 11 y4, 0, 10 y2, 0, 0, 0, 5 y1, 5 y2, 5 y3, 0, 5 y5, 5 y4]

  p = s 2 + s 3 - s 5 - s 6   p = - s 2 + s 4 + s 5 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y6 + y4, y6 - y4 + y2, y2, y3, y6, 0, y5, y6, y4]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8


 
1626 . Coloring, {2, 4, 5, 6, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, 2, 4, 5]
B: [6, 7, 8, 6, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 2 y5 - 2 y2, y5, y4, 2 y5 - 2 y2, 0, y1, y2, 0, 0, y3, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y2, 0, 0, 0, 0, y1, y2, y2, y3, y4, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6


 
1627 . Coloring, {2, 4, 5, 6, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, 2, 1, 9]
B: [6, 7, 8, 6, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y5, y4, y4, 0, 0, 0, y2, y3, y4, 0, y1, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y1, y2, y3, y1, y1, 0, y5, 0, y4]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1628 . Coloring, {2, 4, 5, 6, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, C, 4, 9]
B: [6, 7, 8, 6, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 5 vs 7 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 2 y3, y5, 0, 0, y4, y3, y2, 0, y1, 2 y3]

  p' = - s 3 + s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[2 y3, y1, 0, 0, y2, -2 y3 + 2 y5, y6, y3, 0, y4, 0, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1629 . Coloring, {2, 4, 5, 6, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, 2, 4, 5]
B: [6, 7, 8, 6, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y2, y8, y7, 0, y6, y5, 0, 0, y4, y3]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y7, y7, y5, y6, y4, y3]

  p = s 2 - s 8


 
1630 . Coloring, {2, 4, 5, 6, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, 2, 1, 9]
B: [6, 7, 8, 6, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[2 y1, 7 y1 + 7 y6 - 9 y5 - 9 y4 + 7 y3 - 9 y2, 7 y1 + 7 y6 - 9 y5 - 9 y4 + 7 y3 - 9 y2, 0, 0, 0, 2 y6, 2 y5, 2 y4, 0, 2 y3, 2 y2]

  p = - s 3 + s 5 + s 6 - s 8   p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y4, y3, y3, 0, y5, y6, y7]

  p = s 5 - s 8


 
1631 . Coloring, {2, 4, 5, 6, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, C, 4, 9]
B: [6, 7, 8, 6, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, -14 y5 - 14 y2 + 18 y3 - 14 y4 + 18 y1, 5 y5, 0, 0, 5 y2, -7 y5 - 7 y2 + 9 y3 - 7 y4 + 9 y1, 5 y3, 0, 5 y4, 5 y1]

  p = - s 2 + s 4 + s 5 - s 7   p' = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y4, y5, y6, y4, 0, y3, y7, 0]

  p = s 5 - s 8


 
1632 . Coloring, {2, 4, 5, 6, 10, 11, 12}

R: [7, 8, 7, 7, 3, 3, B, C, B, 2, 4, 9]
B: [6, 7, 8, 6, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y7, y7, y6, 0, 0, y5, y4, y3, 0, y2, y1]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y5, y4, y4, 0, y3, y6, y7]

  p = - s 5 + s 8


 
1633 . Coloring, {2, 4, 5, 7, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, 2, 1, 5]
B: [6, 7, 8, 6, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y1 + y2 - y3 - y4 + y5 + y6 - y7 + y8, y2, 0, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, 0, y1, y2, 0, y3, y6, y4, y5, y6, y7, y8]

  p = - s 2 + s 9


 
1634 . Coloring, {2, 4, 5, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, C, 4, 5]
B: [6, 7, 8, 6, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y6, y5, y4, 0, 0, y3, y2, y1, y10, y9, y8, y7]


 
1635 . Coloring, {2, 4, 5, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, C, 1, 9]
B: [6, 7, 8, 6, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, 0, 0, y7, y5, y6, y4, y2, y3]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, y9, y10, y8, y6, y7, y5, y4, 0, y3, y1, y2]


 
1636 . Coloring, {2, 4, 5, 7, 8, 10, 11}

R: [7, 8, 7, 7, 3, A, A, B, B, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, 0, y5, y6, 0, y7, y8, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y4, 0, y3, 0, 0, y2, y7, y1, y4 + y3 - y2 - y1 - y6 + y5, y7, y6, y5]

  p' = s 7 - s 8   p = s 7 - s 9


 
1637 . Coloring, {2, 4, 5, 7, 8, 10, 12}

R: [7, 8, 7, 7, 3, A, A, B, B, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, 0, 0, y4, y3, 2 y5, y2, y1, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y9, y8]


 
1638 . Coloring, {2, 4, 5, 7, 8, 11, 12}

R: [7, 8, 7, 7, 3, A, A, B, B, C, 4, 9]
B: [6, 7, 8, 6, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y7, y1, 0, 0, y2, y7, y5, y6, y4, y3]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y7, y6, y4, y5, 0, y9, y8, y10]


 
1639 . Coloring, {2, 4, 5, 7, 9, 10, 11}

R: [7, 8, 7, 7, 3, A, A, C, C, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y6, y5, y4, y3, 0, y2, y1, 0, y8, 0, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y2, 0, y1, 0, 0, y8, y7, y6, y5, y7, y4, y3]

  p = s 4 - s 9


 
1640 . Coloring, {2, 4, 5, 7, 9, 10, 12}

R: [7, 8, 7, 7, 3, A, A, C, C, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y5, y4, y5, 0, 0, 0, y1, y2, y3, y7, 0, y6]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, -5 y1 + 11 y2 - 5 y3 - 5 y4 - 5 y5 + 11 y6 - 5 y7 + 11 y8, 5 y1, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1641 . Coloring, {2, 4, 5, 7, 9, 11, 12}

R: [7, 8, 7, 7, 3, A, A, C, C, C, 4, 9]
B: [6, 7, 8, 6, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, 0, y3, y5, y4, y1, 0, y2]

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y9, y8, y7, 0, y6, y5, y4, y3, 0, y2, y1, 0]


 
1642 . Coloring, {2, 4, 5, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, C, B, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y9, 0, 0, y8, y7, y5, y6, y3, y4]

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[3 y1, 0, -3 y1 + 5 y4 - 3 y2 - 3 y3 - 3 y8 + 5 y7 - 3 y6 + 5 y5, 0, 3 y4, 3 y2, 3 y3, 3 y8, 0, 3 y7, 3 y6, 3 y5]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1643 . Coloring, {2, 4, 5, 8, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, y3, y2, y4, 0, y1, y3, 0, y5, y6, y5]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y9, 0, y8, 0, 0, y7, y6, y5, y4, y3, y2, y1]


 
1644 . Coloring, {2, 4, 5, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-3 y5 - 6 y6 - 3 y1 - 3 y7 + 13 y2 - 3 y3 + 13 y4, 3 y5, 3 y6, 0, 0, 0, 3 y1, 3 y7, 3 y2, 3 y6, 3 y3, 3 y4]

  p = - s 4 + s 6 + s 7 - s 9   p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y8, y7, 0, y5, y8, y6]

  p = - s 6 + s 9


 
1645 . Coloring, {2, 4, 5, 8, 9, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, C, 4, 9]
B: [6, 7, 8, 6, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y4, -3 y1 - 6 y4 + 10 y3 - 3 y2, 0, 0, 3 y1, 3 y4, -3 y4 + 3 y3, 3 y4, 3 y2, 3 y3]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y2, y3, 0, y6, y7, y5, y4, 0, y10, y8, y9]


 
1646 . Coloring, {2, 4, 5, 8, 10, 11, 12}

R: [7, 8, 7, 7, 3, A, B, B, B, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y4, y5, y3, 0, 0, y2, y1, 2 y5, y5, y6, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y1, 0, y4, 0, y3, y2, y1, y7, 0, y6, 0, y5]

  p = s 5 - s 8


 
1647 . Coloring, {2, 4, 5, 9, 10, 11, 12}

R: [7, 8, 7, 7, 3, A, B, C, C, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, y4 + y1 + y2 - y6, y4, y4 + 2 y1 + 2 y2 - y3 - y5, 0, 0, y3, y1, y2, y4, y5, y6]

  p = s 4 - s 7   p' = s 4 - s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[y2, 0, y1, 0, y2 + y1 + y8 - y7 + y6 - y5 + y3 - y4, y8, y7, y6, 0, y5, y3, y4]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1648 . Coloring, {2, 4, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, 2, 1, 5]
B: [6, 7, 8, 6, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y4, y3, y5, 0, y2, 0, y1, -y4 + y3 + y2 + y1 - y7 + y6 - 2 y5, 0, y7, y6, y5]

  p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y8, y7, 0, y6, y8, y5, y4, y3, y2, y1]

  p = s 3 - s 9


 
1649 . Coloring, {2, 4, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, C, 4, 5]
B: [6, 7, 8, 6, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y4, y2, y1, 0, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, 0, y6, y4, y5, y10, y7, y8, y9]


 
1650 . Coloring, {2, 4, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, C, 1, 9]
B: [6, 7, 8, 6, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y6, 0, 0, 0, y7, y6, y5, y2, y3, y4]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 2 y5, -5 y5 + 13 y1 - 5 y2 + 13 y3 - 5 y8 - 2 y7, 2 y6, 2 y1, 2 y2, -2 y6 + 33 y1 + 33 y3 - 2 y4 - 13 y5 - 13 y2 - 13 y8, 2 y3, 0, 2 y4, 2 y8, 2 y7]

  p = s + s 3 - s 7 - s 9   p' = s + s 3 - s 7 - s 9


 
1651 . Coloring, {2, 4, 6, 7, 8, 10, 11}

R: [7, 8, 7, 7, A, 3, A, B, B, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 2 y2, 0, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y7, 0, y6, 0, 0, y5, y6, y3, y4, y2, y6 + y3, y1]

  p = s 6 - s 8   p' = s 6 - s 8


 
1652 . Coloring, {2, 4, 6, 7, 8, 10, 12}

R: [7, 8, 7, 7, A, 3, A, B, B, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, 0, 0, y4, y3, 2 y5, y2, y1, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
1653 . Coloring, {2, 4, 6, 7, 8, 11, 12}

R: [7, 8, 7, 7, A, 3, A, B, B, C, 4, 9]
B: [6, 7, 8, 6, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y6, y7, 0, 0, y5, y6, y3, y4, y2, y1]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {1, 2, 6, 7, 10, 11}}    order:   12

See Matrix
 

[-3 y1 + 10 y7 - 3 y6 - 3 y5 + 10 y4 - 3 y3 - 3 y2, 3 y1, 3 y7 + 3 y4 - 3 y8, 0, 3 y7, 3 y6, 3 y5, 3 y4, 0, 3 y3, 3 y2, 3 y8]

  p' = - s - s 3 + s 7 + s 9   p = - s - s 3 + s 7 + s 9


 
1654 . Coloring, {2, 4, 6, 7, 9, 10, 11}

R: [7, 8, 7, 7, A, 3, A, C, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y2, 2 y2, y3, 0, y4, y6, 0, y5, 0, y7]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y8, 0, y7, 0, 0, y6, y7, y5, y4, y3, y2, y1]

  p = s 3 - s 9


 
1655 . Coloring, {2, 4, 6, 7, 9, 10, 12}

R: [7, 8, 7, 7, A, 3, A, C, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y1, y2, 0, 0, 0, y3, y4, y7, y5, 0, y6]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{3, 4, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y9, y8, y6, y7, 0, y5, y4, y3]


 
1656 . Coloring, {2, 4, 6, 7, 9, 11, 12}

R: [7, 8, 7, 7, A, 3, A, C, C, C, 4, 9]
B: [6, 7, 8, 6, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y3, 2 y3, 0, 0, y4, y3, y1, y2, 0, y5]

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y9, y8, y7, 0, y6, y5, y4, y3, 0, y2, y1, 0]


 
1657 . Coloring, {2, 4, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, C, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, 0, y4, y7, y5, y6, y8, y9]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y9, 0, y8, y7, y6, y5, 0, y4, y3, y2]


 
1658 . Coloring, {2, 4, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y8, y2, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[2 y7 + y2 - y1 - y5 - y6 + y4 + y3, 0, y7, 0, 0, y2, y7, y1, y5, y6, y4, y3]

  p = - s 3 + s 9   p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1659 . Coloring, {2, 4, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y6, 3 y5, 3 y4, 0, 0, 0, 3 y3, 3 y2, 3 y1, 3 y4, -3 y6 - 3 y5 - 6 y4 - 3 y3 - 3 y2 + 13 y1 + 13 y7, 3 y7]

  p' = s 4 + s 5 - s 7 - s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, -y5 + y2 + y1 - y3, -y4 + y2 + y1, y4, y5, y3, y2, 0, y1, y3, y2 + y1 - y3]

  p = s 5 - s 9   p' = s 5 - s 8   p' = s 6 - s 8   p' = s 7 - s 8


 
1660 . Coloring, {2, 4, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, C, 4, 9]
B: [6, 7, 8, 6, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 4 vs 8 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y4, 3 y3, 0, 0, 3 y2, 3 y4, 3 y1, 3 y4, 4 y4 - 3 y3 - 3 y2 + 10 y1, 3 y4 + 3 y1]

  p' = s 4 - s 7   p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[5 y6, 5 y5, 5 y4, 0, 5 y3, 5 y2, -5 y6 - 5 y5 + 11 y4 + 11 y3 - 5 y2 + 11 y1 - 5 y9 - 5 y8 + 11 y7, 5 y1, 0, 5 y9, 5 y8, 5 y7]

  p = s 4 + s 5 + s 6 - s 8 - s 9 - s 10


 
1661 . Coloring, {2, 4, 6, 8, 10, 11, 12}

R: [7, 8, 7, 7, A, 3, B, B, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 4 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y4, y2, 0, 0, y6, y3, 2 y4, y4, y5, 0]

  p' = - s 4 + s 7   p = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[2 y2, 0, -y2 + y4 - y1, 0, -2 y2 + y4, y1, y2, y4 - y3, 0, y3, 0, y4]

  p = s 4 - s 6   p' = s 4 - s 5   p' = - s 5 + s 7   p' = - s 5 + s 6


 
1662 . Coloring, {2, 4, 6, 9, 10, 11, 12}

R: [7, 8, 7, 7, A, 3, B, C, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, y2, y3, -y3 - y1 + 2 y2 + 2 y5 - y4, 0, 0, y1, y2 - y3 - y6 + y5, y6, y3, y4, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[-y1 + y5 + y6, 0, y5 + y6 - y2 - y3, 0, y1, y2, y3, -y4 + y5 + y6, 0, y4, y5, y6]

  p' = - s 3 + s 4 - s 7 + s 8   p' = s 2 - s 3 + s 6 - s 7   p = s 2 - s 4 + s 6 - s 8


 
1663 . Coloring, {2, 4, 7, 8, 9, 10, 11}

R: [7, 8, 7, 7, A, A, A, B, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 - y2 - y3 + y7 + y4 - y5 + y6, 0, y1, y2, 0, y3, y7, 0, y4, y5, y6]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y8, 0, y7, 0, 0, y6, y5, y3, y4, 0, y1, y2]


 
1664 . Coloring, {2, 4, 7, 8, 9, 10, 12}

R: [7, 8, 7, 7, A, A, A, B, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y2 + 8 y3 - 3 y4 + 5 y6, -3 y1 + 5 y3 - 3 y5 + 8 y6, 0, 0, 0, 0, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s + s 7   p' = - s + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y2 + y3 - y4 + y5 + y6 - y7, y1, y2, y3, y4, y5, 0, 0, y6, y7]

  p = - s 5 + s 6 - s 7 + s 8


 
1665 . Coloring, {2, 4, 7, 8, 9, 11, 12}

R: [7, 8, 7, 7, A, A, A, B, C, C, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, 0, 0, y6, y5, y3, y4, y2, y1]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y3, y1, y2, 0, y4, y5, y6, y7, 0, 0, y8, y9]


 
1666 . Coloring, {2, 4, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, B, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, y4, y5, y7, y6, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, 0, y7, y8]


 
1667 . Coloring, {2, 4, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, A, A, C, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y2, 0, y3, 0, 0, y1, y5, y6, y7, 0, y4]

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y5, y6, y7, 0, 0, y4, 2 y6]

  p = - s 3 + s 8


 
1668 . Coloring, {2, 4, 8, 9, 10, 11, 12}

R: [7, 8, 7, 7, A, A, B, B, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y5, 0, 3 y4, 0, 0, 3 y3, 3 y2, 3 y1, -3 y5 - 3 y4 - 3 y3 - 3 y2 + 13 y1 - 3 y7 + 13 y6, 3 y7, 3 y6]

  p = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y6, 0, y4, 0, y5, y3, y2, y4 - y5 - y3 + y1, 0, y6, y2, y1]

  p' = - s 4 + s 8   p' = s 4 - s 5 + s 6 - s 7   p = s 4 - s 8


 
1669 . Coloring, {2, 5, 6, 7, 8, 9, 10}

R: [7, 8, 7, 6, 3, 3, A, B, C, 2, 1, 5]
B: [6, 7, 8, 7, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1 - y2 + y3 + y4 - y5 - y6 + y8, y1, y2, 0, y3, y7, y4, y5, 0, y6, y8, y7]

  p = - s 4 + s 10   p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y3, y2, y3, y4, y5, y6, y7]

  p = - s 5 + s 8


 
1670 . Coloring, {2, 5, 6, 7, 8, 9, 11}

R: [7, 8, 7, 6, 3, 3, A, B, C, C, 4, 5]
B: [6, 7, 8, 7, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y7, y8, y9, y6, y5, y3, y4]


 
1671 . Coloring, {2, 5, 6, 7, 8, 9, 12}

R: [7, 8, 7, 6, 3, 3, A, B, C, C, 1, 9]
B: [6, 7, 8, 7, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, y3 + y6, 0, 0, y3, y2, y3, y4, y5, y6, y7]

  p' = s 6 - s 8   p = - s 6 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y8, y7, y6, y5, y6, 0, y4, y3, y2]

  p = s 6 - s 9


 
1672 . Coloring, {2, 5, 6, 7, 8, 10, 11}

R: [7, 8, 7, 6, 3, 3, A, B, B, 2, 4, 5]
B: [6, 7, 8, 7, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y4, y5, y3, y2, y1, y9, y8, 0, y7, y6, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 2 y5, y5, y3, y6, y4, y7]

  p = - s 6 + s 8


 
1673 . Coloring, {2, 5, 6, 7, 8, 10, 12}

R: [7, 8, 7, 6, 3, 3, A, B, B, 2, 1, 9]
B: [6, 7, 8, 7, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 4 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y8, y7, y6, 0, 0, y5, y4, y3, 2 y5, y2, y1, 0]

  p = s 3 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y4, y2, y1, y4, y1, 0, 5 y4 - y2 - 2 y1 - y3, y4, y3]

  p = - s 2 + s 8   p' = - s 3 + s 6   p' = - s 2 + s 5   p = - s 2 + s 5


 
1674 . Coloring, {2, 5, 6, 7, 8, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, B, B, C, 4, 9]
B: [6, 7, 8, 7, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y9, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y8, y7, 0, 0, y6, y5, y4, y3, 0, y2, y1, -y8 + y7 + y6 + y5 - y4 + y3 - y2 + y1]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1675 . Coloring, {2, 5, 6, 7, 9, 10, 11}

R: [7, 8, 7, 6, 3, 3, A, C, C, 2, 4, 5]
B: [6, 7, 8, 7, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y8, y9, y7, y5, y6, y4, y3, 0, y2, 0, y1]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 2 y3, y3, y4, y5, y7, y6]

  p = s 2 - s 8


 
1676 . Coloring, {2, 5, 6, 7, 9, 10, 12}

R: [7, 8, 7, 6, 3, 3, A, C, C, 2, 1, 9]
B: [6, 7, 8, 7, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 4 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[2 y7, y1, y8, 0, 0, y7, y6, y5, y4, y3, 0, y2]

  p = s 7 - s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, -27 y4 - 20 y3 + 7 y2 + 16 y1, 2 y4, 2 y3, 2 y2, 2 y3, 0, -16 y4 - 12 y3 + 4 y2 + 10 y1, 2 y1, -7 y4 - 4 y3 + 3 y2 + 4 y1]

  p' = s 4 - s 7   p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 8


 
1677 . Coloring, {2, 5, 6, 7, 9, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, C, C, C, 4, 9]
B: [6, 7, 8, 7, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y4, 0, y2, y3, y4, y6, y5, 0, y7]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 2 y5, y3, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8


 
1678 . Coloring, {2, 5, 6, 7, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, C, B, 2, 4, 9]
B: [6, 7, 8, 7, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 9, 10, 11, 12}}    order:   10

See Matrix
 

[0, y10, y9, y8, 0, y7, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y3, 0, 0, 0, y2, y1, 2 y6, y6, 0, y4, y5, y7]

  p = s 5 - s 8


 
1679 . Coloring, {2, 5, 6, 8, 9, 10, 11}

R: [7, 8, 7, 6, 3, 3, B, B, C, 2, 4, 5]
B: [6, 7, 8, 7, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 2 y7, y1, y2, y3, y4, y5, 2 y3 - 3 y7, 0, 0, y6, y7]

  p' = s 3 - s 8   p = s 3 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 2 y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8


 
1680 . Coloring, {2, 5, 6, 8, 9, 10, 12}

R: [7, 8, 7, 6, 3, 3, B, B, C, 2, 1, 9]
B: [6, 7, 8, 7, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y3, 6 y1, 3 y2, 0, 0, 3 y1, -3 y3 - 9 y2 + 13 y6 - 3 y5 + 13 y4, 6 y2 - 9 y1, 3 y6, 0, 3 y5, 3 y4]

  p = - s 3 + s 9   p' = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 5 + s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, y4, y5, y4, 0, y3, y4, y6]

  p = s 4 - s 7   p' = s 4 - s 7


 
1681 . Coloring, {2, 5, 6, 8, 9, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, B, C, C, 4, 9]
B: [6, 7, 8, 7, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -5 y1 - 5 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 5 y1, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7]

  p = - s 2 - s 3 + s 7 + s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6 + y5, y4, 0, 0, y3, -2 y6 + y5 + y3, y2, y6, 0, y1, y6, y5]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7


 
1682 . Coloring, {2, 5, 6, 8, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, B, B, 2, 4, 9]
B: [6, 7, 8, 7, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y6, y5, y4, 0, y3, y2, y1, y6, 0, y7, 0]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, y2, 2 y3, y3, 0, y4, 0, y5]

  p' = s 3 - s 6   p = s 3 - s 6


 
1683 . Coloring, {2, 5, 6, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, C, C, 2, 4, 9]
B: [6, 7, 8, 7, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 7 y1, 9 y1 - 7 y8 - 7 y7 - 7 y6 + 9 y5 + 9 y4 - 7 y3 + 9 y2, 7 y8, 0, 7 y7, 7 y6, 7 y5, 7 y4, 0, 7 y3, 7 y2]

  p = s 3 + s 4 - s 8 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 2 y4, y4, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1684 . Coloring, {2, 5, 7, 8, 9, 10, 11}

R: [7, 8, 7, 6, 3, A, A, B, C, 2, 4, 5]
B: [6, 7, 8, 7, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y2 - y5 - y3 - y4 + y6 + y7 - y8 + y9, y1, y2, y5, y3, y4, y6, 0, y7, y8, y9]

  p = - s 5 + s 6 - s 7 + s 8 - s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y2, 0, y1, 0, 0, y3, 2 y5, y6, y7, y5, y4, y8]

  p = - s 2 + s 9


 
1685 . Coloring, {2, 5, 7, 8, 9, 10, 12}

R: [7, 8, 7, 6, 3, A, A, B, C, 2, 1, 9]
B: [6, 7, 8, 7, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 6 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, -3 y7 - 3 y5 - 3 y3 + 8 y2 + 5 y1, -3 y6 - 3 y5 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = - s 3 + s 9   p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 2 y1, 4 y6 - 16 y5 + 10 y4 + 10 y3 + 4 y2, -2 y1 + 7 y6 - 27 y5 + 16 y4 + 16 y3 + 7 y2, 2 y6, 2 y5, 2 y4, 0, 2 y3, 3 y6 - 7 y5 + 4 y4 + 4 y3 + 3 y2, 2 y2]

  p' = - s 5 + s 8   p' = - s 4 + s 7   p = s 4 - s 7


 
1686 . Coloring, {2, 5, 7, 8, 9, 11, 12}

R: [7, 8, 7, 6, 3, A, A, B, C, C, 4, 9]
B: [6, 7, 8, 7, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y3, y1, 0, y2, y3 + y6, y3, y4, y5, y6, y7]

  p = - s 6 + s 8   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y4, y5, y7, y6, 0, y8, y9, y10]


 
1687 . Coloring, {2, 5, 7, 8, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, A, B, B, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y6, y7, y8, 0, y5, y4, y1, 2 y7, y3, y2, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y2, 0, y1, 0, y8, y7, y6, y5, 0, y4, y3, y9]


 
1688 . Coloring, {2, 5, 7, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, A, C, C, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y3, 2 y3, 0, -3 y3 + 2 y2, y2, y5, y6, y7, 0, y4]

  p = s 6 - s 8   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[5 y6, 0, 5 y5, 0, 5 y4, 5 y3, 5 y2, -5 y6 - 5 y5 + 11 y4 - 5 y3 - 5 y2 + 11 y1 - 5 y8 + 11 y7, 0, 5 y1, 5 y8, 5 y7]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1689 . Coloring, {2, 5, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, B, B, C, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y4, 3 y3, 3 y2, 0, -3 y4 - 3 y3 + 8 y8 - 3 y6 + 5 y5, 3 y1, -3 y2 - 3 y1 + 5 y8 - 3 y7 + 8 y5, 3 y8, 3 y7, 3 y6, 3 y5]

  p' = - s 3 + s 9   p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y6, y7, 2 y3, y8, 0, y2, y3, y4]

  p = - s 6 + s 9


 
1690 . Coloring, {2, 6, 7, 8, 9, 10, 11}

R: [7, 8, 7, 6, A, 3, A, B, C, 2, 4, 5]
B: [6, 7, 8, 7, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y5, y4, y3, y2, y1, -y5 + y4 + y3 - y2 - y1 + y6 + y9 - y8 + y7, y6, 0, y9, y8, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y7 + y6 - y5 - y4 - y3 + y2 + y1, 0, y7, 0, 0, y6, 2 y7, y5, y4, y3, y2, y1]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = - s 3 + s 9


 
1691 . Coloring, {2, 6, 7, 8, 9, 10, 12}

R: [7, 8, 7, 6, A, 3, A, B, C, 2, 1, 9]
B: [6, 7, 8, 7, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y8 - 3 y7 + 5 y6 - 3 y4 + 8 y2, -3 y1 - 3 y5 + 8 y6 - 3 y3 + 5 y2, 0, 0, 3 y8, 3 y7, 3 y5, 3 y6, 3 y3, 3 y4, 3 y2]

  p' = - s 3 + s 9   p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 0, 9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y6 + 9 y7 - 7 y8, 7 y1, 7 y2, 7 y3, 7 y4, 7 y5, 0, 7 y6, 7 y7, 7 y8]

  p = s 3 + s 4 + s 5 - s 7 - s 8 - s 9


 
1692 . Coloring, {2, 6, 7, 8, 9, 11, 12}

R: [7, 8, 7, 6, A, 3, A, B, C, C, 4, 9]
B: [6, 7, 8, 7, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y5, y4, 0, y3, y2, y1, y9, y8, y7, y6]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 3 y4, 3 y3, 3 y2, 5 y7 - 8 y5 - 3 y4 + 5 y2 + 5 y1 - 8 y8, 0, 3 y1, 8 y7 - 3 y6 - 11 y5 - 3 y3 + 8 y2 + 8 y1 - 11 y8, 3 y8]

  p' = - s - s 3 + s 7 + s 9   p = - s - s 3 + s 7 + s 9


 
1693 . Coloring, {2, 6, 7, 8, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, A, B, B, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, y9, y4, y5, y8, y6, y7, 0]

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y3, 0, y2, 0, y1, y3 - y2 + y1 + y8 + y7 + y6 - y5 - y4, y8, y7, 0, y6, y5, y4]

  p = s 6 - s 7 + s 8 - s 9


 
1694 . Coloring, {2, 6, 7, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, A, C, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1, y2, y9, 0, y7, y8, y5, y6, y4, 0, y3]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y8 - y4 - y5 + y6 + y7, 0, y1, 0, y2, y3, y8, y4, 0, y5, y6, y7]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1695 . Coloring, {2, 6, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, B, B, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 5 vs 9

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y8 - 3 y9 - 3 y5 - 3 y6 + 13 y7 - 3 y2 - 3 y3 + 13 y4, 3 y1, 3 y8, 0, 3 y9, 3 y5, 3 y6, 3 y7, 3 y2, 3 y3, 3 y4]

  p = s 4 + s 5 - s 9 - s 10

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y5, 0, y3, 0, y4, y5 - y3 + y4 - 2 y1, 2 y1, y2, 0, y5 + y4 - y2, y1, y5 + y4 - y1]

  p = s 5 - s 9   p' = s 5 - s 8   p' = s 7 - s 8   p' = s 6 - s 8


 
1696 . Coloring, {2, 7, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, A, A, B, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y2 - 3 y1 + 5 y7 - 3 y5 + 8 y4, 0, 3 y3, 0, 3 y2, 3 y1, -3 y3 + 8 y7 - 3 y6 + 5 y4, 3 y7, 3 y6, 3 y5, 3 y4]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, y6, 0, 0, y7, y8]


 
1697 . Coloring, {3, 4, 5, 6, 7, 8, 9}

R: [7, 7, 8, 7, 3, 3, A, B, C, C, 1, 5]
B: [6, 8, 7, 6, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y3, 0, y2, 0, y8, y7, 0, y6, y5, y4]

Omega Rank for B :  cycles: {{2, 4, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y2 - y1 + y3 + y4 + y5 + y7 - y8 - y6, 0, y2, 0, y1, y3, y4, y5, y7, y8, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1698 . Coloring, {3, 4, 5, 6, 7, 8, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, 2, 1, 5]
B: [6, 8, 7, 6, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y4, 0, y5, y6, 0, y8, y7, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y1 + 2 y6 + y5 + y4 - y3 - y2, y6, y6, y5, y4, y3, y2]

  p' = s 6 - s 7   p = s 6 - s 8


 
1699 . Coloring, {3, 4, 5, 6, 7, 8, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, C, 4, 5]
B: [6, 8, 7, 6, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, y1 - y3 + y4 + y2 + y7 + y8 - y6 - y5, 0, 0, 0, y3, y4, y2, y7, y8, y6, y5]

  p = s 8 - s 9


 
1700 . Coloring, {3, 4, 5, 6, 7, 8, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, C, 1, 9]
B: [6, 8, 7, 6, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, 0, y4, y2, y3, y5, y6, 0, y9, y7, y8]


 
1701 . Coloring, {3, 4, 5, 6, 7, 9, 10}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, 2, 1, 5]
B: [6, 8, 7, 6, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[7 y7, 7 y6, 7 y5, 0, 7 y4, 0, 7 y3, 7 y2, 0, -7 y7 - 7 y6 + 9 y5 + 9 y4 - 7 y3 + 9 y2 + 9 y1, 0, 7 y1]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, 0, y5, y1, y1, -y4 + y5 - 2 y1 - y2 + y3 + y6, y2, y3, y6]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = - s 2 + s 8


 
1702 . Coloring, {3, 4, 5, 6, 7, 9, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, C, 4, 5]
B: [6, 8, 7, 6, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 7 vs 7 6 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, y6, y5, y4, 0, y3, y2, 0, y1, 0, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y6 - 3 y5 - y4 - y2 + y3, y1, 0, 0, 0, y6, y5, y4, 2 y5, y2, y3, 0]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1703 . Coloring, {3, 4, 5, 6, 7, 9, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, C, 1, 9]
B: [6, 8, 7, 6, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y5 - y4, 0, y5 - y4, 0, 0, 0, y5, y4, y3, y2, 0, y1]

  p' = - s 4 + s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 2 y4, y3, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8


 
1704 . Coloring, {3, 4, 5, 6, 7, 10, 11}

R: [7, 7, 8, 7, 3, 3, A, C, B, 2, 4, 5]
B: [6, 8, 7, 6, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 3 y2, 3 y1, -3 y2 + 10 y1 + 10 y3 - 3 y7 - 3 y5 - 3 y4, 3 y1 - 3 y6 + 3 y3, 0, 3 y7, 3 y6, 0, 3 y5, 3 y4, 3 y3]

  p = s 3 + s 5 - s 6 - s 8   p' = s 3 + s 5 - s 6 - s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y4 - 2 y3 - y2 - y1 + y5 + y6, 0, 0, 0, 0, y4, y3, y3, y2, y1, y5, y6]

  p = s 6 - s 8   p' = s 6 - s 7


 
1705 . Coloring, {3, 4, 5, 6, 7, 10, 12}

R: [7, 7, 8, 7, 3, 3, A, C, B, 2, 1, 9]
B: [6, 8, 7, 6, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y3, y2, 0, 0, 0, y9, y7, y6, y8, y5, y4]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, y2, y1, y5, y5, 0, y4, y7, y6]

  p = - s 5 + s 8


 
1706 . Coloring, {3, 4, 5, 6, 7, 11, 12}

R: [7, 7, 8, 7, 3, 3, A, C, B, C, 4, 9]
B: [6, 8, 7, 6, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, 0, 0, y3, y4, y2, y1, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y4, y5, y8, y3, 0, y6, y7, y8]

  p = s 3 - s 9


 
1707 . Coloring, {3, 4, 5, 6, 8, 9, 10}

R: [7, 7, 8, 7, 3, 3, B, B, C, 2, 1, 5]
B: [6, 8, 7, 6, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y7, 2 y1, y6, 0, y5, 0, y4, y3, 0, 0, y2, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y1, y3, y3, y4, y5, y7, y6]

  p = s 2 - s 8


 
1708 . Coloring, {3, 4, 5, 6, 8, 9, 11}

R: [7, 7, 8, 7, 3, 3, B, B, C, C, 4, 5]
B: [6, 8, 7, 6, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y7, y5, y6, 0, y3, y4, 0, 0, y2, y1]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y5, y9, y7, y8, y6]


 
1709 . Coloring, {3, 4, 5, 6, 8, 9, 12}

R: [7, 7, 8, 7, 3, 3, B, B, C, C, 1, 9]
B: [6, 8, 7, 6, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 7, 11}}    order:   6

See Matrix
 

[-5 y1 - 5 y2 - 5 y3 + 11 y4 - 5 y5 + 11 y6, 0, 5 y1, 0, 0, 0, 5 y2, 5 y3, 5 y4, 0, 5 y5, 5 y6]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y8, 0, y6, y7, y5, y3, y4, 0, y2, y3, y1]

  p = - s 4 + s 9


 
1710 . Coloring, {3, 4, 5, 6, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, 2, 4, 5]
B: [6, 8, 7, 6, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, y5, y4, y6, 0, y3, y2, 0, 0, y1, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[2 y4, 0, 0, 0, 0, y5, y4, y4, y3, y2, 0, y1]

  p' = - s 4 + s 6   p = s 4 - s 6


 
1711 . Coloring, {3, 4, 5, 6, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, 2, 1, 9]
B: [6, 8, 7, 6, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y4, y4, 0, 0, 0, y2, y3, y4, 0, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 2 y3, y1, y2, y3, y3, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1712 . Coloring, {3, 4, 5, 6, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, C, 4, 9]
B: [6, 8, 7, 6, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 9 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, -2 y3 + 2 y4, y1, 0, 0, y2, y3, y4, 0, y5, -2 y3 + 2 y4]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y4, y7, 0, 0, y6, y5, y4, y3, 0, y2, 0, y1]

  p = - s 3 + s 8


 
1713 . Coloring, {3, 4, 5, 6, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, 2, 4, 5]
B: [6, 8, 7, 6, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}, {4, 7, 11}}    order:   12

See Matrix
 

[0, 9 y1 - 7 y2 + 9 y3 - 7 y4 + 9 y5 - 7 y6 + 9 y7, 7 y1, 7 y2, 7 y3, 0, 7 y4, 7 y5, 0, 0, 7 y6, 7 y7]

  p = s 2 + s 3 + s 4 - s 6 - s 7 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8


 
1714 . Coloring, {3, 4, 5, 6, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, 2, 1, 9]
B: [6, 8, 7, 6, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y1, 7 y1 + 7 y4 - 9 y3 - 9 y2 + 7 y5 - 9 y6, 7 y1 + 7 y4 - 9 y3 - 9 y2 + 7 y5 - 9 y6, 0, 0, 0, 2 y4, 2 y3, 2 y2, 0, 2 y5, 2 y6]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, y5, y4, y3, y3, 0, y2, y1, y7]

  p = s 5 - s 8


 
1715 . Coloring, {3, 4, 5, 6, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, C, 4, 9]
B: [6, 8, 7, 6, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 9 y1 + 9 y5 - 7 y4 - 7 y3 + 9 y2 - 7 y6, 7 y1, 0, 0, 7 y5, 7 y4, 7 y3, 0, 7 y2, 7 y6]

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y4, y3, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8


 
1716 . Coloring, {3, 4, 5, 6, 10, 11, 12}

R: [7, 7, 8, 7, 3, 3, B, C, B, 2, 4, 9]
B: [6, 8, 7, 6, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y1, y2, 0, 0, y3, y4, y5, 0, y6, y7]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y5, y6, y7, y7, 0, y2, y3, y4]

  p = - s 5 + s 8


 
1717 . Coloring, {3, 4, 5, 7, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, 2, 1, 5]
B: [6, 8, 7, 6, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y1, y3, y2, 0, y4, 0, y6, y5, 0, y9, y8, y7]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y4, y3, y6, y5, y6, y8, y7]

  p = s 4 - s 9


 
1718 . Coloring, {3, 4, 5, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, C, 4, 5]
B: [6, 8, 7, 6, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y2, y1, -y1 + y4 + y3, 0, -y2 + y4 + y3, -y5 + y4 + y3, 0, y5, y4, y3]

  p' = - s 2 + s 3 - s 6 + s 7   p' = s - s 2 + s 5 - s 6   p = s - s 3 + s 5 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   10

See Matrix
 

[y1, y2, y3, 0, 0, y4, y5, y6, y7, y8, y9, y10]


 
1719 . Coloring, {3, 4, 5, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, C, 1, 9]
B: [6, 8, 7, 6, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 + y4 - y2 - y3 - y5 + y6 + y7, 0, y1, 0, 0, 0, y4, y2, y3, y5, y6, y7]

  p = s 7 - s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y4, y3, y4 + y3 - 2 y2 + 3 y1 - 2 y5, -y4 - 7 y3 + 13 y2 - 8 y1 + 6 y5, 3 y4 + 8 y3 - 14 y2 + 9 y1 - 7 y5, -2 y4 - 6 y3 + 11 y2 - 7 y1 + 7 y5, y2, 0, y1, -2 y4 - 13 y3 + 23 y2 - 14 y1 + 11 y5, y5]

  p' = - s 4 + s 9   p' = - s + s 6   p' = - s 3 + s 8   p' = - s 2 + s 7   p = - s + s 6


 
1720 . Coloring, {3, 4, 5, 7, 8, 10, 11}

R: [7, 7, 8, 7, 3, A, A, B, B, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 6 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y4, y2, y3, 0, y7, y5, 0, y6, y8, 0]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[y2, 0, y1, 0, 0, -y2 - y1 - y3 + 2 y6 - y5, y3, y4, -2 y4 + y6, y4, y5, y6]

  p' = s 2 - s 7   p' = s 3 - s 8   p = s 2 - s 7


 
1721 . Coloring, {3, 4, 5, 7, 8, 10, 12}

R: [7, 7, 8, 7, 3, A, A, B, B, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, 0, 0, y5, y4, 2 y2, y6, y7, 0]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, y5, y6, y2, y3, y4, y5 + y6 - y2 + y3 + y4 - y1 + y7 - y8, 0, y1, y7, y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1722 . Coloring, {3, 4, 5, 7, 8, 11, 12}

R: [7, 7, 8, 7, 3, A, A, B, B, C, 4, 9]
B: [6, 8, 7, 6, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, 0, y2, y3, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[-y4 + y2 + y1, y5 + y4 - y2, y4 - y2 - y1 + y5 + y3, 0, y5 - y2 + y3, y5, y4, y3, 0, y2, y1, -y3 + y2 + y1]

  p' = - s + s 6   p' = - s 3 + s 8   p' = - s 2 + s 7   p = - s + s 6   p' = - s 4 + s 9


 
1723 . Coloring, {3, 4, 5, 7, 9, 10, 11}

R: [7, 7, 8, 7, 3, A, A, C, C, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -3 y1 + 10 y2 - 3 y3 + 10 y4 - 3 y5, 3 y2 + 3 y4 - 3 y6, 3 y1, 3 y2, 0, 3 y3, 3 y4, 0, 3 y5, 0, 3 y6]

  p' = - s 2 - s 4 + s 5 + s 7   p = s 2 + s 4 - s 5 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, 0, y3, 0, 0, y8, y1, y7, y5, y7, y6, y4]

  p = - s 4 + s 9


 
1724 . Coloring, {3, 4, 5, 7, 9, 10, 12}

R: [7, 7, 8, 7, 3, A, A, C, C, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[6 y2 + 6 y3 - 6 y5, 4 y2 + 4 y3 + 6 y5 - 3 y1 - 3 y4, 3 y2 + 3 y3 - 3 y5, 0, 0, 0, 3 y1, 3 y2, 3 y3, 3 y4, 0, 3 y5 ]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 5 y2, 5 y3, 5 y4, 5 y1, -5 y2 - 5 y3 + 11 y4 - 5 y1 - 5 y8 + 11 y7 - 5 y5 + 11 y6, 5 y8, 0, 5 y7, 5 y5, 5 y6]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1725 . Coloring, {3, 4, 5, 7, 9, 11, 12}

R: [7, 7, 8, 7, 3, A, A, C, C, C, 4, 9]
B: [6, 8, 7, 6, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y3 - 2 y4, 2 y3 - 4 y4, 0, 0, y3, y4, y2, y1, 0, -y4 + y2 + y1]

  p' = - s 5 + s 6   p = s 4 - s 6   p' = s 4 - s 5

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, 0]


 
1726 . Coloring, {3, 4, 5, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, C, B, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y9, y8, y7, 0, 0, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[-3 y8 + 5 y7 - 3 y6 - 3 y5 - 3 y4 + 5 y3 - 3 y2 + 5 y1, 0, 3 y8, 0, 3 y7, 3 y6, 3 y5, 3 y4, 0, 3 y3, 3 y2, 3 y1]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1727 . Coloring, {3, 4, 5, 8, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y1, 0, y7, y5, 0, y6, y4, y6]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, y6, y7, y8, y9, y5, y4]


 
1728 . Coloring, {3, 4, 5, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-9 y3 - 6 y6 - 3 y1 + 13 y5 - 3 y4 + 13 y2, 3 y3 + 3 y6, 3 y3, 0, 0, 0, 3 y1, 3 y6, 3 y5, 3 y3, 3 y4, 3 y2]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, y7, 0, y6, y7, y8]

  p = s 6 - s 9


 
1729 . Coloring, {3, 4, 5, 8, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, C, 4, 9]
B: [6, 8, 7, 6, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 3 y2, -3 y2 - 3 y1 - 3 y5 - 3 y3 + 10 y4, 0, 0, 3 y1, 3 y5, -3 y2 + 3 y4, 3 y2, 3 y3, 3 y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   10

See Matrix
 

[y6, y7, y5, 0, y4, y3, y1, y2, 0, y9, y10, y8]


 
1730 . Coloring, {3, 4, 5, 8, 10, 11, 12}

R: [7, 7, 8, 7, 3, A, B, B, B, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y3 + y4, y3, y1, 0, 0, y5, y4, 2 y3, y3, y2, 0]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y5, 0, y1, 0, y2, y3, y4, y5, 0, y6, 0, y7]

  p = - s 5 + s 8


 
1731 . Coloring, {3, 4, 5, 9, 10, 11, 12}

R: [7, 7, 8, 7, 3, A, B, C, C, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y5, 3 y4, 3 y3, 0, 0, 3 y1, 3 y5 - 3 y4, 3 y2, 3 y4, -3 y3 - 3 y1 + 7 y5 - 13 y4 + 10 y2, 3 y5 - 6 y4 + 3 y2]

  p' = - s 5 + s 8   p' = s 3 - s 6   p' = s 4 - s 7   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y9, 0, y8, 0, y7, y6, y5, y4, 0, y3, y2, y1]


 
1732 . Coloring, {3, 4, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, 2, 1, 5]
B: [6, 8, 7, 6, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, y7, 0, y7 + y4, 0, y3, y4, 0, y5, y6, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y4, y1, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
1733 . Coloring, {3, 4, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, C, 4, 5]
B: [6, 8, 7, 6, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 8 10 vs 10

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y10, y8, y9, 0, 0, y7, y6, y4, y5, y3, y1, y2]


 
1734 . Coloring, {3, 4, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, C, 1, 9]
B: [6, 8, 7, 6, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1 + y7 - y2 - y3 - y4 + y5 + y6, 0, y1, 0, 0, 0, y7, y2, y3, y4, y5, y6]

  p = - s 7 + s 8

Omega Rank for B :  cycles: {{2, 3, 4, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[0, y6, y7, y1, y2, y3, y4, y5, 0, y8, y9, y10]


 
1735 . Coloring, {3, 4, 6, 7, 8, 10, 11}

R: [7, 7, 8, 7, A, 3, A, B, B, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, 2 y2, 0, y4, y5, 0, y6, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y8, 0, y7, 0, 0, y5, y6, y7, y3, y4, y1, y2]

  p = - s 7 + s 9


 
1736 . Coloring, {3, 4, 6, 7, 8, 10, 12}

R: [7, 7, 8, 7, A, 3, A, B, B, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y3, 0, 0, 0, y4, y6, 2 y3, y5, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y9, y8, y7, y6, y5, y4, 0, y3, y2, y1]


 
1737 . Coloring, {3, 4, 6, 7, 8, 11, 12}

R: [7, 7, 8, 7, A, 3, A, B, B, C, 4, 9]
B: [6, 8, 7, 6, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y2 + y3 + y4 + y5 - y6 - y7, y1, 0, 0, y2, y3, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y10, y9, y8, 0, y7, y6, y5, y4, 0, y3, y2, y1]


 
1738 . Coloring, {3, 4, 6, 7, 9, 10, 11}

R: [7, 7, 8, 7, A, 3, A, C, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y1, 2 y1, y3, 0, y4, y5, 0, y6, 0, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y4, 0, 0, y2, y3, y4, y5, y6, y7, y8]

  p = - s 3 + s 9


 
1739 . Coloring, {3, 4, 6, 7, 9, 10, 12}

R: [7, 7, 8, 7, A, 3, A, C, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[6 y3 + 6 y2 - 6 y5, -3 y1 + 4 y3 + 4 y2 - 3 y4 + 6 y5, 3 y3 + 3 y2 - 3 y5, 0, 0, 0, 3 y1, 3 y3, 3 y2, 3 y4, 0, 3 y5 ]

  p = s 3 - s 6   p' = s 3 - s 6   p' = s 4 - s 7

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4, y5, y9, y6, y7, y8, 0, y1, y2, y3]


 
1740 . Coloring, {3, 4, 6, 7, 9, 11, 12}

R: [7, 7, 8, 7, A, 3, A, C, C, C, 4, 9]
B: [6, 8, 7, 6, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 4 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1 - 2 y2 - 2 y3 + 2 y4, 2 y1 - 4 y2 - 4 y3 + 4 y4, 0, 0, y1, y2 + y3 - y4, y2, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y9, 0, y7, y8, 0]


 
1741 . Coloring, {3, 4, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, C, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y9, y8, 0, 0, y7, y6, y5, y4, y3, y2]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y6, 0, y4, y5, y3, y2, 0, y8, y9, y7]


 
1742 . Coloring, {3, 4, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5, y6, y4, y3, 0, y2, -y6 + y3, 0, y1, y7, y6]

  p = s 5 - s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y8, 0, y7, 0, 0, y6, y5, y7, y4, y3, y2, y1]

  p = s 3 - s 9


 
1743 . Coloring, {3, 4, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-9 y2 - 3 y5 - 6 y4 + 13 y3 - 3 y1 + 13 y6, 3 y2 + 3 y4, 3 y2, 0, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, 3 y1, 3 y6]

  p = - s 3 + s 9   p = s 3 - s 5 - s 6 + s 8   p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y2, y3, y8, y7, y6, y5, 0, y4, y5, y1]

  p = s 4 - s 9


 
1744 . Coloring, {3, 4, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, C, 4, 9]
B: [6, 8, 7, 6, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 5 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y2, 3 y3, 0, 0, 3 y1, -3 y3 - 3 y1 - 3 y2 - 3 y5 + 10 y4, -3 y2 + 3 y4, 3 y2, 3 y5, 3 y4]

  p = - s 3 + s 6   p' = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[y9, y10, y7, 0, y8, y6, y5, y4, 0, y2, y3, y1]


 
1745 . Coloring, {3, 4, 6, 8, 10, 11, 12}

R: [7, 7, 8, 7, A, 3, B, B, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y5, y4, y3, 0, 0, y2, y5 - y4, 2 y4, y4, y1, 0]

  p' = - s 3 + s 6   p = - s 3 + s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[2 y2, 0, y6, 0, y5, y4, y3, y2, 0, y1, 0, y7]

  p = - s 3 + s 8


 
1746 . Coloring, {3, 4, 6, 9, 10, 11, 12}

R: [7, 7, 8, 7, A, 3, B, C, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1 + 3 y3, 3 y1, 4 y1 - 3 y3 - 3 y2 + 10 y5 - 3 y4, 0, 0, 3 y2, 3 y3, 3 y1 - 3 y3 + 3 y5, 3 y1, 3 y4, 3 y5]

  p' = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9   p = - s 3 + s 6

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y9, 0, y7, y8, y4, y5, 0, y6, y2, y3]


 
1747 . Coloring, {3, 4, 7, 8, 9, 10, 11}

R: [7, 7, 8, 7, A, A, A, B, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, y5 + y6, 0, y3, y5, 0, y4, y6, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y5, 0, y4, 0, 0, y3, y2, y1, y6, 0, y7, y8]


 
1748 . Coloring, {3, 4, 7, 8, 9, 10, 12}

R: [7, 7, 8, 7, A, A, A, B, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y2, 3 y3, 0, 0, 0, 0, -3 y2 - 3 y3 - 3 y1 + 13 y4 - 3 y5 - 3 y6 + 13 y7, 3 y1, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y8, y7, y6, y5, y4, y3, 0, 0, y2, y1]


 
1749 . Coloring, {3, 4, 7, 8, 9, 11, 12}

R: [7, 7, 8, 7, A, A, A, B, C, C, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, 0, y2, y6, y7, y5, y3, y4]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y5, y6, y7, 0, y8, y9, y3, y4, 0, 0, y2, y1]


 
1750 . Coloring, {3, 4, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, B, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, 0, y5, 0, 0, y4, y3, 2 y3, y2, y1, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y4, 0, y3, 0, y1, y2, y8, y7, 0, 0, y5, y6]


 
1751 . Coloring, {3, 4, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, A, A, C, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, 10 y1 + 10 y5 - 22 y4 + 10 y3 - 22 y2, 0, 0, y5, 5 y1 + 5 y5 - 11 y4 + 5 y3 - 11 y2, y4, y3, 0, y2]

  p = - s 2 + s 4 + s 5 - s 7   p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y7, y4, y5, y6, 0, 0, y3, 2 y6]

  p = s 3 - s 8


 
1752 . Coloring, {3, 4, 8, 9, 10, 11, 12}

R: [7, 7, 8, 7, A, A, B, B, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1, 0, -3 y1 - 3 y4 - 9 y5 + 13 y3 - 3 y2 + 13 y6, 0, 0, 3 y4, 3 y5, 3 y3, 6 y5, 3 y2, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y8, y7, y6, y5, 0, y4, y5, y3]

  p = s 4 - s 9


 
1753 . Coloring, {3, 5, 6, 7, 8, 9, 10}

R: [7, 7, 8, 6, 3, 3, A, B, C, 2, 1, 5]
B: [6, 8, 7, 7, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y6, y5, y4, 0, y3, y7, y2, y1, 0, y9, y8, y7]

  p = s 7 - s 10

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y1, 0, y6, y7, y6, y4, y5, y3, y2]

  p = - s 5 + s 8


 
1754 . Coloring, {3, 5, 6, 7, 8, 9, 11}

R: [7, 7, 8, 6, 3, 3, A, B, C, C, 4, 5]
B: [6, 8, 7, 7, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y6, y5, y4, y3, y2, y1, 0, y9, y8, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y2, 0, 0, 0, y3, y4, y9, y5, y6, y7, y8]


 
1755 . Coloring, {3, 5, 6, 7, 8, 9, 12}

R: [7, 7, 8, 6, 3, 3, A, B, C, C, 1, 9]
B: [6, 8, 7, 7, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y3, 0, y2, 0, 0, y1, y3 - y2 + y1 + y8 + y6 + y7 - y5 - y4, y8, y6, y7, y5, y4]

  p = s 8 - s 9

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}, {4, 7, 11}}   

See Matrix
 

[0, 9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y6 + 9 y7 - 7 y8, 0, 7 y1, 7 y2, 7 y3, 7 y4, 7 y5, 0, 7 y6, 7 y7, 7 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1756 . Coloring, {3, 5, 6, 7, 8, 10, 11}

R: [7, 7, 8, 6, 3, 3, A, B, B, 2, 4, 5]
B: [6, 8, 7, 7, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, y2, 5 y2 - y5 - y3 - y4 - y1 - y6, y5, y3, y4, y2, y1, 0, y2, y6, 0]

  p' = - s 2 + s 7   p' = - s 3 + s 8   p = - s 2 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y3, 2 y2, y2, y4, y7, y6, y5]

  p = s 6 - s 8


 
1757 . Coloring, {3, 5, 6, 7, 8, 10, 12}

R: [7, 7, 8, 6, 3, 3, A, B, B, 2, 1, 9]
B: [6, 8, 7, 7, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y7, y8, 0, 0, y6, y4, y5, 2 y6, y2, y3, 0]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y3, 5 y3 - 2 y1 - y2 - y4, y1, y3, y1, 0, y2, y3, y4]

  p' = s 2 - s 5   p = - s 2 + s 5   p' = - s 3 + s 6   p = - s 2 + s 8


 
1758 . Coloring, {3, 5, 6, 7, 8, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, B, B, C, 4, 9]
B: [6, 8, 7, 7, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y8, y9, y3, y4, 0, y5, y6, y7]


 
1759 . Coloring, {3, 5, 6, 7, 9, 10, 11}

R: [7, 7, 8, 6, 3, 3, A, C, C, 2, 4, 5]
B: [6, 8, 7, 7, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, y6, y5, y4, y3, y2, y6, y1, 0, y6, 0, -y5 - y4 - y3 - y2 + 5 y6 - y1]

  p' = s 4 - s 8   p' = s 3 - s 7   p = s 3 - s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, 2 y2, y2, y3, y4, y5, y6]

  p = - s 2 + s 8


 
1760 . Coloring, {3, 5, 6, 7, 9, 10, 12}

R: [7, 7, 8, 6, 3, 3, A, C, C, 2, 1, 9]
B: [6, 8, 7, 7, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[2 y6, 2 y4 + 2 y3 - y5 - y2, y6 + y4 + y3 - y1, 0, 0, y6, y5, y4, y3, y2, 0, y1]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, -27 y1 - 20 y2 + 7 y3 + 16 y4, 2 y1, 2 y2, 2 y3, 2 y2, 0, -16 y1 - 12 y2 + 4 y3 + 10 y4, 2 y4, -7 y1 - 4 y2 + 3 y3 + 4 y4]

  p = s 2 - s 8   p' = s 2 - s 5   p' = s 3 - s 6   p' = s 4 - s 7


 
1761 . Coloring, {3, 5, 6, 7, 9, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, C, C, C, 4, 9]
B: [6, 8, 7, 7, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, -2 y2 + 2 y5, 0, y2, -2 y2 + 2 y5, y3, y4, y5, 0, y6]

  p = - s 5 + s 7   p' = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y6, y5, y6, y7, 0, y3, y4, 0]

  p = - s 2 + s 8


 
1762 . Coloring, {3, 5, 6, 7, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, C, B, 2, 4, 9]
B: [6, 8, 7, 7, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 9, 11, 12}}   

See Matrix
 

[0, y7, y6, y5, 0, y4, y7, y3, y2, y7, y1, 5 y7 - y6 - y5 - y4 - y3 - y2 - y1]

  p' = - s 2 + s 9   p' = - s + s 8   p = - s + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 2 y4, y4, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1763 . Coloring, {3, 5, 6, 8, 9, 10, 11}

R: [7, 7, 8, 6, 3, 3, B, B, C, 2, 4, 5]
B: [6, 8, 7, 7, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y7, y1, y2, y5, y4, 2 y5 - 2 y7, y3, 0, 0, y6, y7]

  p = - s 3 + s 8   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y6, 2 y5, y5, y4, y3, y2, y1]

  p = s 2 - s 8


 
1764 . Coloring, {3, 5, 6, 8, 9, 10, 12}

R: [7, 7, 8, 6, 3, 3, B, B, C, 2, 1, 9]
B: [6, 8, 7, 7, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y2, 6 y1, -3 y2 - 9 y1 - 3 y7 - 3 y5 + 13 y6 - 3 y3 + 13 y4, 0, 0, 3 y1, 3 y7, 3 y5, 3 y6, 0, 3 y3, 3 y4]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, y2, y4, y1, y4, 0, y6, y4, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1765 . Coloring, {3, 5, 6, 8, 9, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, B, C, C, 4, 9]
B: [6, 8, 7, 7, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 5 y7, 5 y6, 0, 5 y5, 5 y4, 5 y3, 5 y2, 0, -5 y7 - 5 y6 - 5 y5 - 5 y4 - 5 y3 + 11 y2 + 11 y1, 5 y1]

  p = s 2 + s 3 - s 7 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y6, y5, 0, 0, y4, y3, 2 y7, y2, 0, y1, y7, y8]

  p = - s 4 + s 9


 
1766 . Coloring, {3, 5, 6, 8, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, B, B, 2, 4, 9]
B: [6, 8, 7, 7, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 7 vs 8 5 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y6, y1, y2, 0, y3, y4, y5, y6, 0, y7, 0]

  p = - s 3 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y2, 0, 0, 0, y3, y1, 2 y2, y2, 0, y4, 0, y5]

  p = s 3 - s 6   p' = s 3 - s 6


 
1767 . Coloring, {3, 5, 6, 9, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, C, C, 2, 4, 9]
B: [6, 8, 7, 7, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y1 + y2 - y3 + y4 - y5 - y6 - y8 + y7, y1, y2, 0, y3, y4, y5, y6, 0, y8, y7]

  p = - s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y6, y5, 2 y4, y4, 0, y3, y2, y1]

  p = - s 5 + s 8


 
1768 . Coloring, {3, 5, 7, 8, 9, 10, 11}

R: [7, 7, 8, 6, 3, A, A, B, C, 2, 4, 5]
B: [6, 8, 7, 7, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y7, y8, y6, y4, y5, y3, y2, 0, y1, y9, y10]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y8, 0, y7, 0, 0, y5, y6, y4, y3, y4, y2, y1]

  p = s 4 - s 9


 
1769 . Coloring, {3, 5, 7, 8, 9, 10, 12}

R: [7, 7, 8, 6, 3, A, A, B, C, 2, 1, 9]
B: [6, 8, 7, 7, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 5 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 6 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8 ]

  p = - s 5 - s 6 + s 8 + s 9   p = s 5 - s 7 - s 8 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 2 y1, 2 y2, 3 y1 + 3 y2 + 4 y5 + 4 y4 - 7 y3, 2 y5, 7 y1 + 7 y2 + 16 y5 + 16 y4 - 27 y3, 2 y5, 0, 4 y1 + 4 y2 + 8 y5 + 10 y4 - 16 y3, 2 y4, 2 y3]

  p = - s 3 + s 6   p' = - s 4 + s 7   p = - s 3 + s 9   p' = - s 3 + s 6


 
1770 . Coloring, {3, 5, 7, 8, 9, 11, 12}

R: [7, 7, 8, 6, 3, A, A, B, C, C, 4, 9]
B: [6, 8, 7, 7, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, 3 y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y2, 2 y1, y3, y4, y5, y6, y7]

  p = - s 7 + s 8   p = - s 7 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[2 y2, 6 y2 + 8 y1 - 12 y3 + 4 y5 - 8 y4, 2 y1, 0, 8 y2 + 13 y1 - 18 y3 + 8 y5 - 15 y4, -4 y2 - 8 y1 + 14 y3 - 8 y5 + 12 y4, 8 y2 + 15 y1 - 22 y3 + 12 y5 - 17 y4, 2 y3, 0, 2 y5, 12 y2 + 18 y1 - 28 y3 + 14 y5 - 22 y4, 2 y4]

  p' = - s 4 + s 9   p' = - s 3 + s 8   p' = - s 2 + s 7   p = - s + s 6   p' = - s + s 6


 
1771 . Coloring, {3, 5, 7, 8, 10, 11, 12}

R: [7, 7, 8, 6, 3, A, A, B, B, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y6, y5, y4, 0, y3, y2, y1, 2 y5, y8, y7, 0]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y2, 0, y1, -y7 - y6 + y5 - y4 - y3 + y2 + y1 + y8, y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1772 . Coloring, {3, 5, 7, 9, 10, 11, 12}

R: [7, 7, 8, 6, 3, A, A, C, C, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, y2, -6 y5 - 4 y3 + 3 y2 + 3 y1 + 3 y4, -12 y5 - 8 y3 + 6 y2 + 6 y1 + 6 y4, 0, -8 y5 - 2 y3 + 3 y2 + 3 y1 + 3 y4, y1, 3 y2 - 7 y5 - 3 y3 + 3 y1 + 3 y4, y5, y4, 0, y3]

  p = - s 3 + s 6   p = - s 3 + s 9   p' = s 4 - s 7   p' = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[5 y6, 0, 5 y5, 0, 5 y4, 5 y3, 5 y2, -5 y6 - 5 y5 + 11 y4 - 5 y3 - 5 y2 + 11 y1 - 5 y8 + 11 y7, 0, 5 y1, 5 y8, 5 y7]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1773 . Coloring, {3, 5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 5s 4 - 8s 5 + 8s 7 - 16s 8

R: [7, 7, 8, 6, 3, A, B, B, C, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 + 8 y8 - 3 y6 + 5 y7, 3 y1, -3 y3 - 3 y4 + 5 y8 - 3 y5 + 8 y7, 0, 3 y2, 3 y3, 3 y4, 3 y8, 3 y5, 3 y6, 3 y7]

  p' = - s 3 + s 9   p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, y5, 0, y4, y2, y3, y8, 0, y1, y8, y7]

  p = s 6 - s 9


 
1774 . Coloring, {3, 6, 7, 8, 9, 10, 11}

R: [7, 7, 8, 6, A, 3, A, B, C, 2, 4, 5]
B: [6, 8, 7, 7, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, 5 y1 + 5 y2 - 3 y3 + 5 y4 - 3 y5 + 5 y6 - 3 y7 + 5 y8 - 3 y9, 3 y1, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 0, 3 y7, 3 y8, 3 y9]

  p = - s 3 - s 4 - s 5 + s 8 + s 9 + s 10

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 - y2 - y4 - y7 + y5 + y6, 0, y3, 0, 0, y1, y2, y3, y4, y7, y5, y6]

  p = - s 3 + s 9   p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1775 . Coloring, {3, 6, 7, 8, 9, 10, 12}

R: [7, 7, 8, 6, A, 3, A, B, C, 2, 1, 9]
B: [6, 8, 7, 7, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[-3 y1 - 3 y2 - 3 y3 - 3 y9 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 0, 3 y3, 3 y9, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 6 - s 7 + s 9 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y3, y2, y1, y5, y4, y5, 0, y8, y6, y7]

  p = s 6 - s 9


 
1776 . Coloring, {3, 6, 7, 8, 9, 11, 12}

R: [7, 7, 8, 6, A, 3, A, B, C, C, 4, 9]
B: [6, 8, 7, 7, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 10

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, 5 y6, 5 y7, 0, -5 y6 - 5 y7 - 5 y1 + 6 y2 + 6 y3 - 5 y4, 5 y2 + 5 y3 - 5 y5, 5 y1, 5 y2, 5 y3, 5 y4, 5 y5]

  p' = - s 3 + s 8   p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y1, y2, y1 - y2 + y8 - y7 + y5 + y6 + y3 - y4 - y9, 0, y8, y7, y5, y6, 0, y3, y4, y9]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10


 
1777 . Coloring, {3, 6, 7, 8, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, A, B, B, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, 7 y1, 9 y1 - 7 y2 - 7 y3 + 9 y4 - 7 y5 - 7 y6 + 9 y7 - 7 y8, 7 y2, 0, 7 y3, 7 y4, 7 y5, 7 y6, 7 y7, 7 y8, 0]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y1 + y7 - y4 + y5 + y6 + y3 - y2 - y8, 0, y7, y4, y5, y6, 0, y3, y2, y8]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1778 . Coloring, {3, 6, 7, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, A, C, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, 7 y2, 7 y1, 9 y2 - 7 y1 - 7 y8 + 9 y7 - 7 y6 - 7 y5 + 9 y4 - 7 y3, 0, 7 y8, 7 y7, 7 y6, 7 y5, 7 y4, 0, 7 y3]

  p = s 5 + s 6 - s 8 - s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y4 - y5 - y6 + y7 + y8, 0, y1, 0, y2, y3, y4, y5, 0, y6, y7, y8]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1779 . Coloring, {3, 6, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, B, B, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -3 y3 - 3 y1 - 3 y2 - 3 y4 - 3 y5 + 13 y6 - 3 y7 - 3 y8 + 13 y9, 3 y3, 3 y1, 0, 3 y2, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8, 3 y9]

  p = - s 4 - s 5 + s 9 + s 10

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y2, 0, y1, y2, y8]

  p = s 4 - s 9


 
1780 . Coloring, {3, 7, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, A, A, B, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 0, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, y5, y4, y3, y2, 0, 0, y1, y8]


 
1781 . Coloring, {4, 5, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, 2, 1, 5]
B: [6, 8, 8, 6, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y3, y1, y2, 0, y3, 0, y6, 0, 0, y4, y5, y5]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - y5 - y6 - y2 + y3 + y4, 0, y1, 0, y5, y6, y2, y3, y4]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1782 . Coloring, {4, 5, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, C, 4, 5]
B: [6, 8, 8, 6, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, 0, 0, y5, y6, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1, y1 - y2 + y7 + y5 + y4 - y6 - y3, 0, 0, 0, y2, 0, y7, y5, y4, y6, y3]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1783 . Coloring, {4, 5, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, C, 1, 9]
B: [6, 8, 8, 6, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y4 - y5 - y1 - y2 + y3, 0, 2 y2, 0, 0, 0, y4, 0, y5, y1, y2, y3]

  p = - s 5 + s 6   p = - s 5 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y4, 0, y3, y2, y1, 0, y8, 0, y7, y6, y5]


 
1784 . Coloring, {4, 5, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = 3s 2 - 14s 3 + 24s 5 + 32s 6 + 32s 7 - 128s 8   p' = - 3s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, A, B, B, 2, 4, 5]
B: [6, 8, 8, 6, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y1, y2, y2, y5, 0, y3, 0, 0, y4, y5, 0]

  p = - s 3 + s 6   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1 - y2 - y3 + y5, 0, 0, 0, 0, y1, 0, y4, y2, y3, y4, y5]

  p = s 5 - s 7   p' = s 5 - s 6


 
1785 . Coloring, {4, 5, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 + 8s 5 + 32s 7   p = - 9s 2 + 4s 4 + 24s 5 + 16s 6 + 96s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, A, B, B, 2, 1, 9]
B: [6, 8, 8, 6, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y5, y4, y3, 0, 0, 0, y2, 0, y3, y1, y6, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y5, y6, 0, y3, 0, y1, y3, y2]

  p = - s 4 + s 7


 
1786 . Coloring, {4, 5, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 - 24s 5 + 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, A, B, B, C, 4, 9]
B: [6, 8, 8, 6, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y6, y5, 0, 0, y4, 0, y3, y2, y6 + y5 - y4 + y3 + y2 - y1, y1]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y4, y5, 0, 0, y6, y7, 0, y1, 0, y2, y3, y8]


 
1787 . Coloring, {4, 5, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 + 4s 4 + 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 7, 7, 7, 3, 3, A, C, C, 2, 4, 5]
B: [6, 8, 8, 6, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y4, y3, y5, y2, 0, y1, 0, 0, y6, 0, y5]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6 - y4 - y5 - y1 + y2 + y3, 0, 0, 0, 0, y6, 0, y4, y5, y1, y2, y3]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1788 . Coloring, {4, 5, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, A, C, C, 2, 1, 9]
B: [6, 8, 8, 6, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[y3, y4, y3, 0, 0, 0, y2, 0, y1, -2 y3 - y4 - y2 + 6 y1, 0, y1]

  p = s 2 - s 5   p' = s 3 - s 6   p' = s 2 - s 5

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y3, y2, 0, y1, 0, y7, y6, y5]


 
1789 . Coloring, {4, 5, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, A, C, C, C, 4, 9]
B: [6, 8, 8, 6, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 6 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y4, y4, 0, 0, y3, 0, y2, y1, 0, 2 y4 - y3 + y2 + y1]

  p' = s 4 - s 5   p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y4, 0, y5, 0, y7, y6, 0]


 
1790 . Coloring, {4, 5, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, C, B, 2, 4, 9]
B: [6, 8, 8, 6, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 2 y3, y2, 0, 0, y7, 0, y6, y5, y4, y3]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y3, y2, 0, y5, 0, y4, y7, y6]


 
1791 . Coloring, {4, 5, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, 2, 4, 5]
B: [6, 8, 8, 6, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y5, y1, y2, y4, 0, y3, 0, 0, 0, y6, y5]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, 0, y3, y4, y5, y7, y6]


 
1792 . Coloring, {4, 5, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, 2, 1, 9]
B: [6, 8, 8, 6, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y4, 3 y3, 3 y3, 0, 0, 0, 3 y2, 0, 3 y1, 0, -3 y4 - 6 y3 - 3 y2 + 13 y1 + 13 y5, 3 y5]

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y3, y2, 0, 2 y6, 0, y1, y6, y5]

  p = s 4 - s 7


 
1793 . Coloring, {4, 5, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, C, 4, 9]
B: [6, 8, 8, 6, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 6 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, -5 y1 - 5 y2 + 11 y3 - 5 y4 + 11 y5, 5 y1, 0, 0, 5 y2, 0, 5 y3, 0, 5 y4, 5 y5]

  p = s 2 + s 3 - s 5 - s 6

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y8, y7, 0, 0, y5, y6, 0, y4, 0, y3, y1, y2]


 
1794 . Coloring, {4, 5, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, B, B, B, 2, 4, 9]
B: [6, 8, 8, 6, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 6 5 vs 6

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y3, y3, y1, 0, 0, y2, 0, y3, 0, y4, 0]

  p' = - s 2 + s 5   p = - s 2 + s 5

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[y2, 0, 0, 0, y4, y3, 0, y2, 0, y1, 0, y5]

  p = s 3 - s 6


 
1795 . Coloring, {4, 5, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, B, C, C, 2, 4, 9]
B: [6, 8, 8, 6, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y1, -2 y1 - y2 - y3 + 6 y4, 0, 0, y2, 0, y4, 0, y3, y4]

  p = s 2 - s 5   p' = s 2 - s 5   p' = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 0, y4, 0, y7, y6, y5]


 
1796 . Coloring, {4, 5, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, A, A, B, C, 2, 4, 5]
B: [6, 8, 8, 6, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y4, y5, y6, y6, 0, y1, 0, 0, y2, y3, y3]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y6, 0, y3, y4, y5, y7, y8]


 
1797 . Coloring, {4, 5, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, A, A, B, C, 2, 1, 9]
B: [6, 8, 8, 6, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 6 y3 - 3 y6 + 13 y5 - 3 y4 + 13 y2, 3 y3, 0, 0, 0, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, y4, 0, y5, 0, y6, y7, y8]


 
1798 . Coloring, {4, 5, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, A, A, B, C, C, 4, 9]
B: [6, 8, 8, 6, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y4, y3, 0, 0, y1, 0, y2, -y3 + y1 - y2 + y5, y4, y5]

  p = - s 5 + s 6   p = - s 5 + s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, y5, 0, y3, y4, 0, y6, 0, y7, y8, y9]


 
1799 . Coloring, {4, 5, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, B, B, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y2, y1, y6, 0, 0, y5, 0, 2 y1, y4, y3, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y8, 0, y5, y6, y7]


 
1800 . Coloring, {4, 5, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, A, A, C, C, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, y2, y1, 2 y1, 0, 0, -y2 - 3 y1 + 6 y3 - y4, 0, y3, y4, 0, y3]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 8, 11}}   

See Matrix
 

[-5 y1 + 11 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 0, 5 y1, 0, 5 y2, 5 y3, 0, 5 y4, 0, 5 y5, 5 y6, 5 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1801 . Coloring, {4, 5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = - s 2 + 2s 3 + 8s 4 - 8s 5 + 32s 7   p = s 2 - 6s 3 + 40s 5 - 32s 6 - 32s 7 + 128s 8

R: [7, 7, 7, 7, 3, A, B, B, C, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -6 y4 - 3 y1 - 3 y2 + 13 y3 - 3 y5 + 13 y6, 3 y4, 3 y1, 0, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, y7, y8]


 
1802 . Coloring, {4, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p' = - 3s 2 + 2s 3 + 8s 5 + 32s 7   p = 3s 2 - 2s 3 - 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, A, B, C, 2, 4, 5]
B: [6, 8, 8, 6, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 8 vs 9 5 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[0, y3, y5, y2, y2, 0, y1, 0, 0, y4, y5, y5]

  p = - s 3 + s 6   p' = - s 4 + s 7   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6, 0, y5, 0, 0, y4, 0, y3, y1, y2, y8, y7]


 
1803 . Coloring, {4, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p' = 3s 2 - 2s 3 - 8s 4 - 8s 5 + 32s 7   p = 9s 2 - 28s 4 - 40s 5 - 16s 6 + 96s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, A, B, C, 2, 1, 9]
B: [6, 8, 8, 6, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 6 y4 - 3 y2 + 13 y3 - 3 y5 + 13 y6, 3 y1, 3 y4, 0, 0, 0, 3 y2, 0, 3 y3, 3 y5, 3 y4, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y8, y7, y6, y5, 0, y4, 0, y3, y2, y1]


 
1804 . Coloring, {4, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 2 + 14s 3 - 24s 5 + 32s 6 - 32s 7 - 128s 8   p' = 3s 2 + 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 7, 7, 7, A, 3, A, B, C, C, 4, 9]
B: [6, 8, 8, 6, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 7 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y4, y2 - y1 - y3 + y5, 0, 0, y2, 0, y1, y3, y4, y5]

  p = s 5 - s 7   p' = s 5 - s 6

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y9, y8, y7, 0, y6, y5, 0, y4, 0, y3, y2, y1]


 
1805 . Coloring, {4, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, B, B, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y4, y3, y2, 0, 0, y1, 0, 2 y3, y6, y5, 0]

  p = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y4, 0, y1, 0, y2, y3, 0, y7, 0, y5, y6, y8]


 
1806 . Coloring, {4, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 7, A, 3, A, C, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 4 vs 7 8 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y1 - y2 + 6 y4 - y3, y1, 2 y1, 0, 0, y2, 0, y4, y3, 0, y4]

  p' = - s 3 + s 6   p = - s 2 + s 5   p' = - s 2 + s 5

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, y7, y8]


 
1807 . Coloring, {4, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 7, 7, 7, A, 3, B, B, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -6 y4 - 3 y1 - 3 y2 + 13 y3 - 3 y5 + 13 y6, 3 y4, 3 y1, 0, 0, 3 y2, 0, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p' = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y8, 0, y7, 0, y6, y5, 0, y4, 0, y3, y2, y1]


 
1808 . Coloring, {4, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, A, A, A, B, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y6 - 3 y1 + 13 y2 - 3 y3 - 3 y4 + 13 y5, 0, 3 y6, 0, 0, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5]

  p = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, 0, y6, y7]


 
1809 . Coloring, {5, 6, 7, 8, 9, 10, 11}

R: [7, 7, 7, 6, 3, 3, A, B, C, 2, 4, 5]
B: [6, 8, 8, 7, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y3, y4, y5, 0, 0, y6, y7, y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y6, y5, 2 y5, y4, y3, y2, y1]

  p = - s 2 + s 8


 
1810 . Coloring, {5, 6, 7, 8, 9, 10, 12}

R: [7, 7, 7, 6, 3, 3, A, B, C, 2, 1, 9]
B: [6, 8, 8, 7, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 9 4 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y1, -6 y1 - 6 y5 - 3 y2 + 13 y6 - 3 y4 + 13 y3, 3 y1, 0, 0, 3 y5, 3 y2, 0, 3 y6, 3 y4, 3 y5, 3 y3]

  p = s 3 + s 4 - s 6 - s 7   p = - s 3 + s 5 + s 6 - s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, -18 y3 + 14 y2 + 9 y1 + 2 y4, y3, -5 y3 + 4 y2 + 2 y1 + y4, y2, -10 y3 + 8 y2 + 4 y1 + 2 y4, 0, y1, y4, -16 y3 + 13 y2 + 8 y1 + 2 y4]

  p = s 2 - s 5   p' = - s 4 + s 7   p' = - s 3 + s 6   p' = - s 2 + s 5


 
1811 . Coloring, {5, 6, 7, 8, 9, 11, 12}

R: [7, 7, 7, 6, 3, 3, A, B, C, C, 4, 9]
B: [6, 8, 8, 7, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, -y1 + y6 + y7 - y5 - y2 + y3 + y4, y1, 0, y6, y7, 0, y5, y2, y3, y4]

  p = s 7 - s 8

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
1812 . Coloring, {5, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 6s 2 + s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 7, 6, 3, 3, A, B, B, 2, 4, 9]
B: [6, 8, 8, 7, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y4, y5, y6, 0, y7, y8, 0, y1, y2, y3, 0]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1813 . Coloring, {5, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + 3s 3 - 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, 3, A, C, C, 2, 4, 9]
B: [6, 8, 8, 7, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -y1 - y2 - y3 - y4 + 6 y6 - y5, y1, y2, 0, y3, y4, 0, y6, y5, 0, y6]

  p = - s 4 + s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y2, y3, 2 y3, 0, y5, y6, y7]

  p = s 5 - s 8

 ยป SYNC'D !RANK'D


 
1814 . Coloring, {5, 6, 8, 9, 10, 11, 12}

R: [7, 7, 7, 6, 3, 3, B, B, C, 2, 4, 9]
B: [6, 8, 8, 7, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y7 + 13 y4 - 3 y5 + 13 y6, 3 y1, 3 y2, 0, 3 y3, 3 y7, 0, 3 y4, 0, 3 y5, 3 y6]

  p = - s 2 - s 3 + s 7 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y5, y3, 2 y3, 0, y2, y3, y6]

  p' = s 4 - s 7   p = s 4 - s 7


 
1815 . Coloring, {5, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 7, 6, 3, A, A, B, C, 2, 4, 9]
B: [6, 8, 8, 7, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 3 y5, 3 y6, 3 y7, 0, 3 y3, 3 y4, 0, 3 y2, -3 y5 - 6 y6 - 3 y7 - 3 y3 - 3 y4 + 13 y2 + 13 y1, 3 y6, 3 y1]

  p = - s 4 - s 5 + s 7 + s 8   p' = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y1, 0, y9, 0, y8, y7, y6, y5, 0, y4, y3, y2]


 
1816 . Coloring, {6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 9s 3 + 2s 4 + 16s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 6, A, 3, A, B, C, 2, 4, 9]
B: [6, 8, 8, 7, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 3 y3, 3 y2, 3 y1, 0, -3 y3 - 3 y2 - 3 y1 - 3 y8 + 13 y7 - 3 y6 - 3 y5 + 13 y4, 3 y8, 0, 3 y7, 3 y6, 3 y5, 3 y4]

  p = s 5 + s 6 - s 8 - s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1 - y2 + y3 - y4 - y5 - y6 + y7 + y8, 0, y1, 0, y2, y3, y4, y5, 0, y6, y7, y8]

  p = - s 6 + s 7 - s 8 + s 9


 
1817 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, C, 1, 5]
B: [6, 7, 7, 6, A, A, B, C, B, 2, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 3, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[-y5 + y2 + y3, 0, y4, 0, y5, 0, -y4 + y2 + y3, y2, 0, y3, y2 + y3 - y1, y1]

  p = - s + s 2 - s 5 + s 6   p = - s + s 3 - s 5 + s 7   p = - s + s 4 - s 5 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y3, 0, y3 + y1 - y2 - y4 - y5 + y7 + y6, 0, y1, y2, 0, y4, y5, y7, y6]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1818 . Coloring, {2, 3, 4, 5, 6, 7, 8, 10}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 - 8s 5 + 32s 7   p = - 9s 2 + 4s 4 - 24s 5 - 16s 6 + 96s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, A, B, B, 2, 1, 5]
B: [6, 7, 7, 6, A, A, B, C, C, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 + y4 - y5 - y6 + y7, y1, y2, 0, y3, 0, y4, y5, 0, y6, y7, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5 - y4 - y3 - y2 + y1 + y6, 0, y5, y4, 0, y3, y2, y1, y6]

  p = s 6 - s 7


 
1819 . Coloring, {2, 3, 4, 5, 6, 7, 8, 11}

Ωp(Δ)=0:     p' = s 7   p' = s 6   p' = s 5   p' = s 4   p' = s 3   p' = s 2   p = s 2

R: [7, 8, 8, 7, 3, 3, A, B, B, C, 4, 5]
B: [6, 7, 7, 6, A, A, B, C, C, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
1 vs 8 1 vs 8 1 vs 8 1 vs 8 1 vs 8

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1, y1, 0, y1, y1, 0, y1, y1, y1]

  p' = s 6 - s 7   p' = s 5 - s 7   p' = s 4 - s 7   p' = s 3 - s 7   p' = s 2 - s 7   p' = s - s 7   p = s - s 8

Omega Rank for B :  cycles: {{9, 12}, {1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y1, 0, 0, 0, y1, y1, 0, y1, y1, y1, y1]

  p' = - s 3 + s 7   p' = - s 3 + s 6   p' = - s 3 + s 4   p' = - s 3 + s 5   p' = s 2 - s 3   p' = s - s 3   p = s - s 4


` See 8-level graph

`

M              \ ;       N

$ [ [0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1] , [1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1] , [0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1] , [0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1] , [0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1] , [1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 2] , [0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1] , [1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 2, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 0, 2] , [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 0] ] $     $ [ [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] , [0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1] , [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0] ] $

τ= 20 , r'= 7/8

R: [7, 8, 8, 7, 3, 3, A, B, B, C, 4, 5]
B: [6, 7, 7, 6, A, A, B, C, C, 2, 1, 9]

Ranges

Action of R on ranges, [[2], [2]]
Action of B on ranges, [[1], [1]]

Cycles:    R , {{3, 4, 5, 7, 8, 10, 11, 12}},   B , {{9, 12}, {1, 2, 6, 7, 10, 11}}

β({1, 2, 6, 7, 9, 10, 11, 12}) = 1/2
β({3, 4, 5, 7, 8, 10, 11, 12}) = 1/2

Partitions
α([{12}, {10}, {11}, {2, 3}, {5, 6}, {1, 4}, {8, 9}, {7}]) = 1/1

b1 = {12} ` , ` b2 = {10} ` , ` b3 = {11} ` , ` b4 = {2, 3} ` , ` b5 = {5, 6} ` , ` b6 = {1, 4} ` , ` b7 = {8, 9} ` , ` b8 = {7}

Action of R and B on the blocks of the partitions: = [2, 8, 7, 5, 1, 3, 4, 6] [7, 5, 8, 2, 6, 3, 1, 4]
with invariant measure [1, 1, 1, 1, 1, 1, 1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-8 partition graph.

`

Right Group
Coloring {2, 3, 4, 5, 6, 7, 8, 11}
Rank8
R,B [7, 8, 8, 7, 3, 3, A, B, B, C, 4, 5], [6, 7, 7, 6, A, A, B, C, C, 2, 1, 9]
π2 [1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2]
u2 [1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1] (dim 1)
wpp [2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1]
π8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
u8 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

 

 
1820 . Coloring, {2, 3, 4, 5, 6, 7, 8, 12}

Ωp(Δ)=0:     p = 3s 2 - 10s 3 - 40s 5 + 32s 6 - 32s 7 + 128s 8   p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, 3, A, B, B, C, 1, 9]
B: [6, 7, 7, 6, A, A, B, C, C, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, 0, y6, -y2 + y1 + y6 - y4 - y5 + y3 + y7, y4, y5, y3, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y2, 0, y1, -y2 + y1 - y7 + y6 + y5 - y4 + y3, y7, y6, 0, 0, y5, y4, y3]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1821 . Coloring, {2, 3, 4, 5, 6, 7, 9, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, A, C, C, 2, 1, 5]
B: [6, 7, 7, 6, A, A, B, B, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y2 + y5 - y3, -y1 + y2 + y5, y1, 0, y2 + y5 - y4, 0, y2, y3, 0, y4, 0, y5]

  p' = s 5 - s 7   p' = s 6 - s 7   p = s 5 - s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y6, 0, y4, y5, 0, y2, y3, y1, y6 - y4 + y5 + y2 + y3 - y1]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1822 . Coloring, {2, 3, 4, 5, 6, 7, 9, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 + 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, 3, A, C, C, C, 4, 5]
B: [6, 7, 7, 6, A, A, B, B, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, 0, -y2 + y4, y4 - y1, y1, 0, y2, y4 - y3, 0, y3, 0, y4]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y4 - y5 - y6 - y3 + y2, y1, 0, 0, 0, y4, y5, 0, y6, y3, y2, 0]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1823 . Coloring, {2, 3, 4, 5, 6, 7, 9, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, 3, A, C, C, C, 1, 9]
B: [6, 7, 7, 6, A, A, B, B, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y1, 0, y1, 0, 0, 0, y4, y4, y3, y2, 0, y3 + y2]

  p = - s 4 + s 5   p = - s 4 + s 6   p = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y7, 0, y6, y5, y4, y3, 0, 0, y2, y1, 0]


 
1824 . Coloring, {2, 3, 4, 5, 6, 7, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, 2, 4, 5]
B: [6, 7, 7, 6, A, A, B, B, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y2, y1, y2 + y1 - y8 + y7 - y6 - y5 + y4 + y3, y8, 0, y7, y6, 0, y5, y4, y3]

  p = s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1 - y2 - y4 - y5 + y6 + y3, 0, 0, 0, 0, y1, y2, 0, y4, y5, y6, y3]

  p = s 6 - s 7


 
1825 . Coloring, {2, 3, 4, 5, 6, 7, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, 2, 1, 9]
B: [6, 7, 7, 6, A, A, B, B, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y8, y7, y6, 0, 0, 0, y5, y4, y3, y2, y1, y8 - y7 - y6 - y5 + y4 + y3 + y2 - y1]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y6, y5, y4, 0, 0, y3, y2, y1]


 
1826 . Coloring, {2, 3, 4, 5, 6, 7, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, C, 4, 9]
B: [6, 7, 7, 6, A, A, B, B, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y2 + y3 + y4 + y5 - y6 - y7, y1, 0, 0, y2, y3, y4, y5, y6, y7]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y4, y5, 0, 0, y6, y7, y8]


 
1827 . Coloring, {2, 3, 4, 5, 6, 8, 9, 10}

Ωp(Δ)=0:     p' = s 3 + 8s 5 + 8s 6 + 16s 7   p = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, C, 2, 1, 5]
B: [6, 7, 7, 6, A, A, A, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   6

See Matrix
 

[y1, 2 y7, y2, 0, y3, 0, y4, y5, 0, 0, y6, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2 + y1 - y6 - y4 + y5 + y3, 0, y2, y1, 0, y6, y4, y5, y3]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1828 . Coloring, {2, 3, 4, 5, 6, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, B, B, C, C, 4, 5]
B: [6, 7, 7, 6, A, A, A, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y1, y6, y7, 0, y5, y4, 0, 0, y3, y2]

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 0, y5, y6, 0, y3, y4, y7, y8]


 
1829 . Coloring, {2, 3, 4, 5, 6, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, B, B, C, C, 1, 9]
B: [6, 7, 7, 6, A, A, A, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[5 y2, 0, 5 y1, 0, 0, 0, -5 y2 - 5 y1 - 5 y6 + 11 y5 - 5 y4 + 11 y3, 5 y6, 5 y5, 0, 5 y4, 5 y3]

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, 0, y2, y2, y6, y5, 0, 0, y3, y4, y4]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1830 . Coloring, {2, 3, 4, 5, 6, 8, 10, 11}

Ωp(Δ)=0:     p = s 2 + 56s 5 + 16s 6 - 64s 7 - 256s 8   p' = s 2 - 16s 4 - 8s 5 + 16s 6 + 64s 7   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, B, 2, 4, 5]
B: [6, 7, 7, 6, A, A, A, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 6 vs 7 4 vs 6

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y5, y1, y2, y5, 0, y3, y4, 0, 0, y6, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{9, 12}}    order:   4

See Matrix
 

[y1, 0, 0, 0, 0, y2 + y3 - y4, y1, 0, y2, y3, 0, y4]

  p = - s 4 + s 6   p = - s 4 + s 5


 
1831 . Coloring, {2, 3, 4, 5, 6, 8, 10, 12}

Ωp(Δ)=0:     p = s 2 + 8s 5 + 48s 6 + 128s 8   p' = s 2 - 8s 4 - 24s 5 - 16s 6 - 64s 7   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, B, 2, 1, 9]
B: [6, 7, 7, 6, A, A, A, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{1, 7, 11}}    order:   3

See Matrix
 

[y1, y3, y3, 0, 0, 0, y4, y5, y3, 0, y2, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, y3, y1, y2, y3, 0, 0, y4, 0, y5]

  p = - s 3 + s 6


 
1832 . Coloring, {2, 3, 4, 5, 6, 8, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 + 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, B, B, B, C, 4, 9]
B: [6, 7, 7, 6, A, A, A, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, 0, y5, y2, 0, 0, y4, y3, y3, 0, y1, y5]

  p = s 3 - s 6   p' = s 3 - s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   3

See Matrix
 

[y5, y3, 0, 0, y1, y1, y2, 0, 0, y4, 0, y5]

  p' = - s 3 + s 6   p = - s 3 + s 6


 
1833 . Coloring, {2, 3, 4, 5, 6, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 - 8s 5 - 80s 6 - 64s 7 - 128s 8   p' = s 2 - 8s 4 - 8s 5 - 16s 6   p' = s 3 - 8s 5 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, C, C, 2, 4, 5]
B: [6, 7, 7, 6, A, A, A, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, -y1 + 5 y4 - y2 - y3 - y5, y1, y4, y2, 0, y4, y3, 0, 0, y4, y5]

  p = - s 2 + s 6   p' = - s 2 + s 6   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 + y2 - y3 - y6 + y4 + y5, 0, 0, 0, 0, y1, y2, 0, y3, y6, y4, y5]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1834 . Coloring, {2, 3, 4, 5, 6, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 + 8s 5 + 16s 6   p' = s 2 + 8s 5 + 16s 6   p' = s 3 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, 3, B, C, C, 2, 1, 9]
B: [6, 7, 7, 6, A, A, A, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 4 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[y3, y1, y1, 0, 0, 0, y3, 5 y3 - 2 y1 - y2 - y4, y2, 0, y3, y4]

  p' = s 5 - s 7   p' = s 3 - s 7   p' = s 4 - s 6   p = s 3 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y5, 0, 0, y4, y5, y6]

  p = s 4 - s 7


 
1835 . Coloring, {2, 3, 4, 5, 6, 9, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 + 16s 6 + 32s 7 - 64s 8

R: [7, 8, 8, 7, 3, 3, B, C, C, C, 4, 9]
B: [6, 7, 7, 6, A, A, A, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 5 y4 - y1 - y2 - y3, y4, 0, 0, y4, y1, y2, 0, y4, y3 ]

  p = - s 3 + s 5   p' = - s 3 + s 5   p = - s 3 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y3, 0, 0, y6, y1, y4, 0, 0, y5, y6, 0]

  p = - s 4 + s 7


 
1836 . Coloring, {2, 3, 4, 5, 6, 10, 11, 12}

Ωp(Δ)=0:     p = 9s 3 + 8s 5 - 8s 6 + 16s 7 - 64s 8   p = - 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, B, C, B, 2, 4, 9]
B: [6, 7, 7, 6, A, A, A, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y7, y7, y6, 0, 0, y5, y4, y3, 0, y2, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, y6, y1, 2 y4, 0, 0, y5, y4, y3]

  p = - s 4 + s 7


 
1837 . Coloring, {2, 3, 4, 5, 7, 8, 9, 10}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, 3, A, A, B, C, 2, 1, 5]
B: [6, 7, 7, 6, A, 3, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y9, y8, 0, y7, 0, y6, y5, 0, y4, y3, y2]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y4, y3, 0, y2, y1, 0, y8, y7, y5, y6]


 
1838 . Coloring, {2, 3, 4, 5, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 - 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, A, A, B, C, C, 4, 5]
B: [6, 7, 7, 6, A, 3, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y3, y4, y2, 0, y1, y7, 0, y6, y5, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y3, y6, y2, 0, 0, y1, y7, 0, y6, y4, y5, y4]

  p = - s 3 + s 8   p' = - s 3 + s 8


 
1839 . Coloring, {2, 3, 4, 5, 7, 8, 9, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7   p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, A, A, B, C, C, 1, 9]
B: [6, 7, 7, 6, A, 3, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y4, 0, y5, 0, 0, 0, y1, y2, y3, y6, y7, y8]

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, y4, y5, y6, 0, 0, y7, y8, y9]


 
1840 . Coloring, {2, 3, 4, 5, 7, 8, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y4, y2, y1, 0, y7, y6, 0, y5, y8, 0]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {9, 12}}    order:   10

See Matrix
 

[7 y7, 0, 7 y6, 0, 0, 7 y5, 7 y4, 0, 7 y3, 7 y2, -7 y7 - 7 y6 - 7 y5 - 7 y4 + 9 y3 + 9 y2 + 9 y1, 7 y1]

  p = - s 2 - s 3 + s 7 + s 8


 
1841 . Coloring, {2, 3, 4, 5, 7, 8, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y3, y2, 0, 0, 0, y5, y4, 2 y2, y7, y6, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, -7 y1 + 9 y4 - 7 y2 - 7 y3 + 9 y5 - 7 y7 + 9 y6, 7 y1, 7 y4, 7 y2, 7 y3, 0, 0, 7 y5, 7 y7, 7 y6]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1842 . Coloring, {2, 3, 4, 5, 7, 8, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, C, 4, 9]
B: [6, 7, 7, 6, A, 3, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y8, y7, 0, 0, y6, y5, y4, y3, y2, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y4, y3, y2, 0, y1, y8, y7, 0, 0, y6, y5, y9]


 
1843 . Coloring, {2, 3, 4, 5, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 8

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y6, y5, y4, y3, 0, y2, y1, 0, y8, 0, y7]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y8, 0, y7, 0, 0, y6, y5, 0, y4, y3, y2, y1]


 
1844 . Coloring, {2, 3, 4, 5, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y1, y2, 0, 0, 0, y3, y4, y7, y5, 0, y6]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, 5 y7, 5 y6, 5 y5, 5 y4, 5 y3, 0, 0, 5 y2, -5 y7 - 5 y6 + 11 y5 - 5 y4 - 5 y3 + 11 y2 + 11 y1, 5 y1]

  p = s + s 2 + s 3 - s 6 - s 7 - s 8


 
1845 . Coloring, {2, 3, 4, 5, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, C, 4, 9]
B: [6, 7, 7, 6, A, 3, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y5, 2 y5, 0, 0, -2 y5 + 2 y4, y4, y3, y2, 0, y1]

  p = - s 4 + s 6   p' = s 4 - s 6

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, y3, y1, 0, y4, y5, y6, 0, 0, y7, y8, 0]


 
1846 . Coloring, {2, 3, 4, 5, 7, 10, 11, 12}

Ωp(Δ)=0:     p' = s 3 + 8s 5 - 8s 6 + 16s 7   p' = s 2 + 8s 4 - 8s 5 + 16s 6   p = s 2 - 8s 5 - 48s 6 + 64s 7 - 128s 8

R: [7, 8, 8, 7, 3, A, A, C, B, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y2, y3, 0, 0, y4, y5, y6, y9, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[-3 y1 + 5 y2 - 3 y3 - 3 y4 + 5 y5 - 3 y6 + 5 y7, 0, 3 y1, 0, 3 y2, 3 y3, 3 y4, 0, 0, 3 y5, 3 y6, 3 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1847 . Coloring, {2, 3, 4, 5, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 + 8s 5 - 48s 6 - 64s 7 - 128s 8   p' = s 2 + 8s 4 + 8s 5 + 16s 6   p' = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, A, B, B, C, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y2, y3, y1, y2, 0, y7, y6, 0, y5, y4, y5]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y7, 0, y6, 0, 0, y5, y4, 0, y2, y3, y7 + y6 - y5 - y4 + y2 + y3 - y1, y1]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1848 . Coloring, {2, 3, 4, 5, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 8s 5 - 16s 6   p' = s 3 - 8s 6 - 16s 7   p' = s 2 - 8s 5 - 16s 6

R: [7, 8, 8, 7, 3, A, B, B, C, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, -3 y7 - 3 y4 - 3 y5 + 13 y3 - 6 y6 - 3 y2 + 13 y1, 3 y6, 0, 0, 0, 3 y4, 3 y5, 3 y3, 3 y6, 3 y2, 3 y1]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y8, y7, y6, y4, y5, 0, 0, y3, y2, y1]


 
1849 . Coloring, {2, 3, 4, 5, 8, 9, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, A, B, B, C, C, 4, 9]
B: [6, 7, 7, 6, A, 3, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 3 y2, -3 y2 - 3 y1 - 3 y5 + 10 y3 - 3 y4, 0, 0, 3 y1, 3 y5, -3 y2 + 3 y3, 3 y2, 3 y4, 3 y3]

  p = s 3 - s 6   p' = s 3 - s 6   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y4, y1, 0, y2, y3, y5, 0, 0, y6, y7, y7]

  p' = - s 5 + s 8   p = - s 5 + s 8


 
1850 . Coloring, {2, 3, 4, 5, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 8s 5 - 24s 6 - 16s 7 - 64s 8   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, A, B, B, B, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y3, y6, 0, 0, y5, y4, 2 y3, y3, y2, 0]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y7, 0, y6, y5, y4, 0, 0, y3, 0, y2]


 
1851 . Coloring, {2, 3, 4, 5, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 8s 6 + 16s 7   p' = s 3 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, A, B, C, C, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -2 y1 + 2 y3 + 2 y2, 5 y4 - 2 y3 - 2 y2, 2 y4, 0, 0, 2 y4, 2 y3, 2 y2, 5 y4 - 2 y3 - 2 y2, 2 y4, 2 y1]

  p = - s 4 + s 8   p = - s 4 + s 5   p = - s 4 + s 9   p = - s 4 + s 7   p = - s 4 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y8, y7, y6, 0, 0, y4, y5, y3]


 
1852 . Coloring, {2, 3, 4, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p' = 3s 2 - 2s 3 + 8s 4 + 8s 5 + 32s 7   p = 9s 2 + 20s 4 + 40s 5 + 16s 6 + 96s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, A, B, C, 2, 1, 5]
B: [6, 7, 7, 6, 3, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y7, y6, y5, 0, y4, 0, y3, y2, 0, y1, y7 - y6 - y4 - y3 + y2 + y1, y5]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y3, y2, 0, y1, y8, 0, y7, y6, y5, y4]


 
1853 . Coloring, {2, 3, 4, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = 3s 2 + 5s 3 + 10s 4 - 8s 6 - 32s 7 - 32s 8   p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, A, B, C, C, 4, 5]
B: [6, 7, 7, 6, 3, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y2, y1, y7, 0, 0, y3, y4, 0, y8, y5, y6, y7]

  p = - s 3 + s 9


 
1854 . Coloring, {2, 3, 4, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 + 24s 5 - 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, A, B, C, C, 1, 9]
B: [6, 7, 7, 6, 3, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, 0, y3, y6, y4, y5, y7, y8]

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y6, y3, y4, y5, 0, 0, y7, y8, y9]


 
1855 . Coloring, {2, 3, 4, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = - s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y1, y2, 2 y1, 0, y4, y5, 0, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, 0, y3, y5, 0, y4, y6, y7, y8]


 
1856 . Coloring, {2, 3, 4, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y3, y2, y1, 0, 0, 0, y6, y4, 2 y1, y5, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y8, y7, y6, y5, 0, 0, y4, y3, y2]


 
1857 . Coloring, {2, 3, 4, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, C, 4, 9]
B: [6, 7, 7, 6, 3, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y8, 0, 0, y7, y6, y5, y4, y3, y2]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y8, y9, y7, 0, y6, y5, y4, 0, 0, y2, y3, y1]


 
1858 . Coloring, {2, 3, 4, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y1, y2, 2 y2, y3, 0, y4, y7, 0, y6, 0, y5]

  p = s 3 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y2, 0, y1, 0, 0, y8, y7, 0, y6, y5, y3, y4]


 
1859 . Coloring, {2, 3, 4, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y2, y1, y2, 0, 0, 0, y6, y7, y5, y4, 0, y3]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y4, y2, y3, y8, 0, 0, y5, y6, y7]


 
1860 . Coloring, {2, 3, 4, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, C, 4, 9]
B: [6, 7, 7, 6, 3, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1, 2 y1, 0, 0, -2 y1 + 2 y2, y2, y3, y4, 0, y5]

  p' = s 4 - s 6   p = s 4 - s 6

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y3, y4, y5, 0, y6, y7, y8, 0, 0, y1, y2, 0]


 
1861 . Coloring, {2, 3, 4, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 - 8s 5 + 48s 6 + 128s 8   p' = s 3 - 4s 4 + 8s 5 - 8s 6 + 16s 7   p' = s 2 - 8s 4 + 24s 5 - 16s 6 + 64s 7

R: [7, 8, 8, 7, A, 3, A, C, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y7, y8, y9, 0, 0, y6, y3, y4, y5, y1, y2]

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y7, y8, y6]


 
1862 . Coloring, {2, 3, 4, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p' = s 2 + 2s 3 - 8s 5 - 32s 7   p' = - s 2 + 8s 4 + 24s 5 + 16s 6 + 64s 7   p = s 2 + s 3 + 24s 6 - 16s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, B, B, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y8, y2, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 6 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[-y2 + y1 + y7 - y5 - y6 + y3 + y4, 0, y2, 0, 0, y1, y7, 0, y5, y6, y3, y4]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1863 . Coloring, {2, 3, 4, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p' = s 3 + 4s 4 - 8s 6 - 16s 7   p = s 2 + 56s 5 + 16s 6 - 64s 7 - 256s 8   p' = s 2 - 16s 4 - 8s 5 + 16s 6 + 64s 7

R: [7, 8, 8, 7, A, 3, B, B, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 9 8 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y7 - 6 y5 - 3 y6 - 3 y1 + 13 y2 - 3 y3 + 13 y4, 3 y7, 3 y5, 0, 0, 0, 3 y6, 3 y1, 3 y2, 3 y5, 3 y3, 3 y4]

  p = - s 4 - s 5 + s 7 + s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y5, y1, y2, y3, y4, 0, 0, y8, y6, y7]


 
1864 . Coloring, {2, 3, 4, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, A, 3, B, B, C, C, 4, 9]
B: [6, 7, 7, 6, 3, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 9 8 vs 9 5 vs 8 6 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 3 y2, 7 y2 - 3 y5 - 3 y4 + 10 y3 - 3 y1, 0, 0, 3 y5, 3 y4, 3 y3, 3 y2, 3 y1, 3 y2 + 3 y3]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y4, y6, y5, 0, y4, y3, y2, 0, 0, y1, -y5 + y3, -y5 + y3]

  p' = s 5 - s 8   p' = s 4 - s 7   p = s 4 - s 7

 ยป SYNC'D !RANK'D


 
1865 . Coloring, {2, 3, 4, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 8s 5 + 8s 6 + 16s 7   p' = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, A, 3, B, B, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y6, y5, y4, 0, 0, y2, y3, 2 y5, y5, y1, 0]

  p = s 4 - s 7   p' = s 4 - s 7

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y2, 0, y1, 0, y4, y5, y3, 0, 0, y6, 0, y7]


 
1866 . Coloring, {2, 3, 4, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 9s 3 - 16s 5 - 24s 6 - 16s 7 + 64s 8   p' = 3s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, 3, B, C, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 4 vs 9 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 2 y4, 2 y3, 2 y2, 0, 0, 2 y2, 2 y1, -2 y3 + 5 y2 - 2 y1, 2 y3, 2 y2, -2 y4 - 2 y3 + 5 y2]

  p' = s 6 - s 8   p' = s 5 - s 8   p' = s 4 - s 8   p = s 4 - s 9   p' = s 7 - s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y3, 0, y4, 0, y5, y6, y7, 0, 0, y8, y1, y2]


 
1867 . Coloring, {2, 3, 4, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, 0, y1, y2, 0, y8, y7, 0, y6, y4, y5]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y7, 0, y6, 0, 0, y5, y4, 0, y3, 0, y1, y2]


 
1868 . Coloring, {2, 3, 4, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y1, 11 y1 - 2 y4 + 11 y3 - 39 y2 + 11 y5 - 2 y6, 0, 0, 0, 0, 2 y4, 2 y3, 2 y2, 2 y5, 2 y6, 3 y1 + 3 y3 - 11 y2 + 3 y5]

  p' = - s + s 7   p = - s + s 7

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}}    order:   5

See Matrix
 

[0, 0, y1, y2, y3, y4, y5, 0, 0, 0, y6, y7]


 
1869 . Coloring, {2, 3, 4, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, C, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, 0, 0, y2, y1, y7, y6, y5, y4]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y2, 2 y3, y1, 0, y7, y6, y5, 0, 0, 0, y4, y3]

  p = s 3 - s 8


 
1870 . Coloring, {2, 3, 4, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 5 - 32s 7   p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, A, A, A, B, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, 0, 0, y3, y4, y5, y6, y7, 0]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y5, y3, y4, 0, 0, 0, y6, y7]


 
1871 . Coloring, {2, 3, 4, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 + 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 - 16s 6 - 32s 7 + 64s 8

R: [7, 8, 8, 7, A, A, A, C, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 7 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, 0, y7, 0, 0, y6, y5, y4, y3, 0, y2]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y5, 0, y4, 0, y3, y2, y1, 0, 0, 0, y7, y6]


 
1872 . Coloring, {2, 3, 4, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 16s 5 - 8s 6 + 16s 7 + 64s 8   p' = s 3 + 4s 4 - 8s 6 - 16s 7

R: [7, 8, 8, 7, A, A, B, B, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y7 - 3 y6 + 13 y4 - 3 y5 - 3 y2 + 13 y3, 0, 3 y1, 0, 0, 3 y7, 3 y6, 3 y4, 3 y5, 3 y2, 3 y3]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y4, y5, y3, 0, 0, y6, y7, y8]


 
1873 . Coloring, {2, 3, 5, 6, 7, 8, 9, 10}

R: [7, 8, 8, 6, 3, 3, A, B, C, 2, 1, 5]
B: [6, 7, 7, 7, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y6, y5, y4, 0, y3, y8, y2, y1, 0, y9, y7, y8]

  p = - s 4 + s 10

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 0, y4, 0, y5, y1, 0, y2, y3, y6, y7]


 
1874 . Coloring, {2, 3, 5, 6, 7, 8, 9, 11}

R: [7, 8, 8, 6, 3, 3, A, B, C, C, 4, 5]
B: [6, 7, 7, 7, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y6, y5, y4, y3, y2, y1, 0, y9, y8, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 0, y4, y3, 0, y8, y7, y6, y5]


 
1875 . Coloring, {2, 3, 5, 6, 7, 8, 9, 12}

R: [7, 8, 8, 6, 3, 3, A, B, C, C, 1, 9]
B: [6, 7, 7, 7, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y5, 0, 0, y2, y3, y4, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y7, y4, 0, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1876 . Coloring, {2, 3, 5, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = 6s 2 - 5s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, 2, 4, 5]
B: [6, 7, 7, 7, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 2 y5, y4, y5, y6, 0, y7, y8, 0]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y5, 0, 0, 0, 0, y4, y3, 0, y2, y1, y6, y7]


 
1877 . Coloring, {2, 3, 5, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = 9s 3 - 2s 4 + 16s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, 2, 1, 9]
B: [6, 7, 7, 7, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 4 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y4, y3, y2, 0, 0, y1, y8, y7, 2 y1, y6, y5, 0]

  p = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 4 y4 + 4 y3 + 3 y2 - 7 y1, 2 y4, 2 y3, 10 y4 + 10 y3 + 4 y2 - 16 y1, 0, 0, 2 y2, 2 y1, 16 y4 + 16 y3 - 27 y1 + 7 y2]

  p' = s 3 - s 6   p' = s 2 - s 5   p = s 2 - s 5


 
1878 . Coloring, {2, 3, 5, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = 9s 3 + 2s 4 + 16s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, C, 4, 9]
B: [6, 7, 7, 7, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 0, y7, y6, 0, y4, y5, y3, y1, y2, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y6, y7, y8, 0, 0, y5, y3, y4]


 
1879 . Coloring, {2, 3, 5, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, 2, 4, 5]
B: [6, 7, 7, 7, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 9 7 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   4

See Matrix
 

[0, y3, y2, 2 y5, y1, -3 y5 + 2 y6, y5, y4, 0, y6, 0, y7]

  p = - s 4 + s 8   p' = - s 4 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y7, y6, 0, y5, y4, y3, y2]


 
1880 . Coloring, {2, 3, 5, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, 2, 1, 9]
B: [6, 7, 7, 7, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 7 vs 9 4 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[2 y1, y6, y7, 0, 0, y1, 2 y7 - 3 y1, y2, y3, y4, 0, y5]

  p = - s 6 + s 8   p' = - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, 2 y4, 2 y3, 4 y4 - 2 y3 - 7 y2 + 3 y1, 16 y4 - 27 y2 + 7 y1, 0, 0, 2 y2, 2 y1, 10 y4 - 16 y2 + 4 y1 ]

  p' = s 3 - s 6   p = s 2 - s 5   p' = s 2 - s 5

 ยป SYNC'D !RANK'D


 
1881 . Coloring, {2, 3, 5, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 16s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, C, 4, 9]
B: [6, 7, 7, 7, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y1, 2 y2, 0, -3 y2 + 2 y5, y2, y3, y4, y5, 0, y6]

  p' = - s 5 + s 7   p = - s 5 + s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y5, 0, 0, y4, y2, y3, 0, 0, y7, y6, 0]


 
1882 . Coloring, {2, 3, 5, 6, 7, 10, 11, 12}

R: [7, 8, 8, 6, 3, 3, A, C, B, 2, 4, 9]
B: [6, 7, 7, 7, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 10 vs 10 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, y7, y8, y9, y10]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y6, y5, y4, 0, 0, y3, y2, y1]


 
1883 . Coloring, {2, 3, 5, 6, 8, 9, 10, 11}

R: [7, 8, 8, 6, 3, 3, B, B, C, 2, 4, 5]
B: [6, 7, 7, 7, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y6, y5, y4, y3, y2, y6, y1, 0, 0, y7, y6]

  p' = - s 3 + s 8   p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6 + y5 - y4 - y3 + y2 + y1, 0, 0, 0, 0, y6, y5, 0, y4, y3, y2, y1]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1884 . Coloring, {2, 3, 5, 6, 8, 9, 10, 12}

R: [7, 8, 8, 6, 3, 3, B, B, C, 2, 1, 9]
B: [6, 7, 7, 7, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y1 - 9 y2 - 3 y5 - 3 y3 + 13 y4 - 3 y6 + 13 y7, 6 y2, 3 y1, 0, 0, 3 y2, 3 y5, 3 y3, 3 y4, 0, 3 y6, 3 y7]

  p = s 4 + s 5 - s 7 - s 8   p = - s 4 + s 6 + s 7 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y5, y3, 0, 0, y4, y5, y6]

  p = - s 4 + s 7


 
1885 . Coloring, {2, 3, 5, 6, 8, 9, 11, 12}

R: [7, 8, 8, 6, 3, 3, B, B, C, C, 4, 9]
B: [6, 7, 7, 7, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, -5 y1 - 5 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 5 y1, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7]

  p = s 2 + s 3 - s 7 - s 8

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, 0, 0, y6, y4, y3, 0, 0, y1, y2, y2]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
1886 . Coloring, {2, 3, 5, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, B, B, 2, 4, 9]
B: [6, 7, 7, 7, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 8 5 vs 6

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, 2 y2, y6, y5, 0, y4, y2, y3, 2 y2, 0, y1, 0]

  p = - s 2 + s 7   p' = - s 2 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, 2 y1, 2 y2, 3 y3, 0, 0, 2 y4, 0, 2 y5]

  p = s 3 - s 6

 ยป SYNC'D !RANK'D


 
1887 . Coloring, {2, 3, 5, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, B, C, C, 2, 4, 9]
B: [6, 7, 7, 7, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 2 y4, y1, y2, 0, y3, y4, y5, y6, 0, y7, y8]

  p = s 7 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y1, 0, 0, 0, 2 y2, 2 y3, 3 y4, 0, 0, 2 y6, 2 y4, 2 y5]

  p = - s 4 + s 7


 
1888 . Coloring, {2, 3, 5, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, 2, 4, 5]
B: [6, 7, 7, 7, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 9 vs 10 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, y8, y7, y4, y5, 0, y6, y9, y4]

  p = - s 4 + s 10

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y4, 0, y3, 0, 0, y2, y1, 0, y8, y7, y6, y5]


 
1889 . Coloring, {2, 3, 5, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = - 2s 2 + 5s 3 + 2s 4 - 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, 2, 1, 9]
B: [6, 7, 7, 7, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 10 5 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y5, 3 y6, 3 y7, 0, 0, 3 y7, -3 y6 - 6 y7 + 5 y1 - 3 y3 + 8 y4, -3 y5 + 8 y1 - 3 y2 + 5 y4, 3 y1, 3 y2, 3 y3, 3 y4]

  p' = - s 2 + s 8   p' = - s 3 + s 9   p = - s 2 + s 8

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, y2, y1, -2 y2 - 2 y1 + 5 y3 + 5 y4 - 4 y5, y3, 4 y2 + 4 y1 - 6 y3 - 6 y4 + 5 y5, 0, 0, 5 y2 + 5 y1 - 8 y3 - 8 y4 + 6 y5, y4, y5]

  p = s 3 - s 6   p' = s 3 - s 6   p' = s 4 - s 7


 
1890 . Coloring, {2, 3, 5, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = - 2s 2 - 5s 3 + 2s 4 + 16s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, C, 4, 9]
B: [6, 7, 7, 7, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y8, y6, 0, y7, y8, y4, y5, y1, y2, y3]

  p = s 7 - s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y4, y5, y1, 0, y2, y3, y9, 0, 0, y6, y7, y8]


 
1891 . Coloring, {2, 3, 5, 7, 8, 10, 11, 12}

R: [7, 8, 8, 6, 3, A, A, B, B, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, y6, y7, y5, 0, y4, y7, y3, 2 y7, y2, y1, 0]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[-7 y1 + 9 y5 - 7 y6 - 7 y7 + 9 y2 - 7 y3 + 9 y4, 0, 7 y1, 0, 7 y5, 7 y6, 7 y7, 0, 0, 7 y2, 7 y3, 7 y4]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1892 . Coloring, {2, 3, 5, 7, 9, 10, 11, 12}

R: [7, 8, 8, 6, 3, A, A, C, C, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y3, y2, 2 y2, 0, y1, y2, y5, y6, y7, 0, y4]

  p = - s 6 + s 8   p' = s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[-5 y1 + 11 y2 - 5 y3 - 5 y4 + 11 y5 - 5 y6 + 11 y7, 0, 5 y1, 0, 5 y2, 5 y3, 5 y4, 0, 0, 5 y5, 5 y6, 5 y7]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1893 . Coloring, {2, 3, 5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, B, B, C, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 10 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y7, 3 y5, 3 y6, 0, 3 y4, 3 y5, -3 y5 - 3 y6 + 5 y2 - 3 y3 + 8 y1, 3 y2, 3 y3, -3 y7 - 3 y5 - 3 y4 + 8 y2 + 5 y1, 3 y1]

  p' = - s 3 + s 9   p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y4, 0, y5, 0, y3, y1, y2, 0, 0, y6, y7, y8]


 
1894 . Coloring, {2, 3, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, 2, 4, 5]
B: [6, 7, 7, 7, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y9, y8, y7, y6, y4, y5, y2, 0, y3, y1, y5]

  p = s 5 - s 10

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 + y2 - y3 - y4 - y5 + y6 + y7, 0, y1, 0, 0, y2, y3, 0, y4, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1895 . Coloring, {2, 3, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = 6s 2 + s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, 2, 1, 9]
B: [6, 7, 7, 7, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[2 y1, 2 y8, 2 y7, 0, 0, 2 y6, 2 y4, 2 y5, 2 y2, 2 y3, 11 y1 - 2 y8 - 2 y7 + 11 y6 - 2 y4 + 11 y5 - 39 y2 + 11 y3, 3 y1 + 3 y6 + 3 y5 - 11 y2 + 3 y3]

  p = - s 3 + s 9   p' = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 0, y5, y4, y3, y2, y1, 0, 0, y8, y7, y6]


 
1896 . Coloring, {2, 3, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = - 6s 2 + s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, C, 4, 9]
B: [6, 7, 7, 7, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 7 y2, 7 y1, 0, 7 y6, 9 y2 + 9 y1 + 9 y6 + 9 y3 - 7 y4 - 7 y5 + 9 y8 - 7 y7, 7 y3, 7 y4, 7 y5, 7 y8, 7 y7]

  p = s 3 + s 4 - s 8 - s 9

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 + y4 - y5 - y6 + y7 + y8, y1, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1897 . Coloring, {2, 3, 6, 7, 8, 10, 11, 12}

R: [7, 8, 8, 6, A, 3, A, B, B, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y5, 0, y6, y4, y3, 2 y4, y8, y7, 0]

  p = s 4 - s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y7 - y6 + y5 - y4 - y3 + y2 + y1, 0, y7, 0, y6, y5, y4, 0, 0, y3, y2, y1]

  p = s - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1898 . Coloring, {2, 3, 6, 7, 9, 10, 11, 12}

R: [7, 8, 8, 6, A, 3, A, C, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y4, y5, 8 y4 - 4 y5 - 10 y3, 0, y3, 4 y4 - 2 y5 - 5 y3, y2, y1, 10 y4 - 5 y5 - 12 y3, 0, -3 y4 + 4 y3 + y2 + y1]

  p' = - s 5 + s 8   p' = - s 5 + s 6   p' = - s 5 + s 7   p = s 5 - s 6

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y6, 0, y7, 0, y3, y4, y5, 0, 0, -y6 + y7 - y3 + y4 - y5 + y1 + y2, y1, y2]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1899 . Coloring, {2, 3, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 + s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, B, B, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 3 y8, 3 y7, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2, 3 y4, -3 y8 - 3 y7 - 3 y6 - 3 y5 - 6 y4 - 3 y3 + 13 y2 + 13 y1, 3 y1]

  p = - s 3 + s 5 + s 8 - s 10   p = s 3 + s 4 - s 8 - s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y3, y4, y8, 0, 0, y5, y6, y7]


 
1900 . Coloring, {2, 3, 7, 8, 9, 10, 11, 12}

R: [7, 8, 8, 6, A, A, A, B, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 6, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -2 y3 - 2 y2 - 39 y6 - 2 y7 + 11 y1 + 11 y5 + 11 y4, 0, 2 y1, 0, 2 y3, 2 y2, 2 y5, 2 y6, 2 y4, 2 y7, 3 y1 + 3 y5 - 11 y6 + 3 y4]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y5, y3, y4, 0, 0, 0, y6, y7]


 
1901 . Coloring, {2, 4, 5, 6, 7, 8, 9, 10}

R: [7, 8, 7, 7, 3, 3, A, B, C, 2, 1, 5]
B: [6, 7, 8, 6, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y1 + y4 - y3 - y2 + y6 + y8 - y5 + y7, y4, 0, y3, 0, y2, y6, 0, y8, y5, y7]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, 0, y2 + 2 y1 + y6 + y5 - y4 - y3, y1, y1, y6, y5, y4, y3]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1902 . Coloring, {2, 4, 5, 6, 7, 8, 9, 11}

R: [7, 8, 7, 7, 3, 3, A, B, C, C, 4, 5]
B: [6, 7, 8, 6, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[0, 0, y1, y8, y7, 0, y6, y5, 0, y4, y3, y2]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y8, y7, 0, 0, 0, y6, y4, y5, y3, y2, y1, y8 - y7 - y6 + y4 + y5 + y3 + y2 - y1]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9


 
1903 . Coloring, {2, 4, 5, 6, 7, 8, 9, 12}

R: [7, 8, 7, 7, 3, 3, A, B, C, C, 1, 9]
B: [6, 7, 8, 6, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y3, 0, 2 y2, 0, 0, 0, y1, y2, y7, y5, y6, y4]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y3, y4, y5, y6, 0, y7, y8, y9]


 
1904 . Coloring, {2, 4, 5, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, 2, 4, 5]
B: [6, 7, 8, 6, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y4, y2, y1, 0, y7, y6, 0, y5, y8, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, 0, y4, y5, y5, y1, y2, y3, y6 - y4 + 2 y5 + y1 + y2 - y3]

  p = - s 6 + s 8   p = - s 6 + s 7


 
1905 . Coloring, {2, 4, 5, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, 2, 1, 9]
B: [6, 7, 8, 6, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1, y2, y5, 0, 0, 0, y3, y4, y5, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, y4, y3, y3, 0, y5, y7, y6]

  p = s 5 - s 8


 
1906 . Coloring, {2, 4, 5, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, C, 4, 9]
B: [6, 7, 8, 6, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 2 y4, y6, 0, 0, y5, y4, y1, y2, y3, y7]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y6, y3, y4, y7, 0, y8, y5, y9]


 
1907 . Coloring, {2, 4, 5, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, 2, 4, 5]
B: [6, 7, 8, 6, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 3, 5, 7, 8, 10, 12}}    order:   7

See Matrix
 

[0, y2, y3, y4, y5, 0, y6, y7, 0, y8, 0, y1]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y4 - 2 y3 - y2 - y1 + y6 + y5, 0, 0, 0, 0, y4, y3, y3, y2, y1, y6, y5]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1908 . Coloring, {2, 4, 5, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, 2, 1, 9]
B: [6, 7, 8, 6, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y2, y3, y2, 0, 0, 0, y1, y7, y5, y6, 0, y4]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y2, y3, y4, y4, 0, y5, y6, y7]

  p = s 5 - s 8


 
1909 . Coloring, {2, 4, 5, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, C, 4, 9]
B: [6, 7, 8, 6, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, 2 y2, 2 y2, 0, 0, y1, y2, y3, y5, 0, y4]

  p = - s 4 + s 6   p' = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, 2 y6, y7, y5, y6, 0, y3, y4, 0]

  p = - s 2 + s 8


 
1910 . Coloring, {2, 4, 5, 6, 7, 10, 11, 12}

R: [7, 8, 7, 7, 3, 3, A, C, B, 2, 4, 9]
B: [6, 7, 8, 6, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, y1, y3, y2, 0, 0, y5, y4, y7, y6, y9, y8]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y5, y4, y3, y3, 0, y2, y6, y7]

  p = s 5 - s 8


 
1911 . Coloring, {2, 4, 5, 6, 8, 9, 10, 11}

R: [7, 8, 7, 7, 3, 3, B, B, C, 2, 4, 5]
B: [6, 7, 8, 6, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y6, y5, y4, y3, 0, y2, 2 y3 - 3 y6, 0, 0, y1, y6]

  p' = s 4 - s 7   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y2, y3, y3, y4, y5, y6, y7]

  p = - s 2 + s 8


 
1912 . Coloring, {2, 4, 5, 6, 8, 9, 10, 12}

R: [7, 8, 7, 7, 3, 3, B, B, C, 2, 1, 9]
B: [6, 7, 8, 6, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-6 y4 - 3 y3 - 3 y2 + 13 y1 - 3 y6 + 13 y5, 3 y4, 3 y4, 0, 0, 0, 3 y3, 3 y2, 3 y1, 0, 3 y6, 3 y5]

  p' = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 5 + s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, y2, y1, y5, y5, 0, y4, y5, y6]

  p' = s 4 - s 7   p = s 4 - s 7


 
1913 . Coloring, {2, 4, 5, 6, 8, 9, 11, 12}

R: [7, 8, 7, 7, 3, 3, B, B, C, C, 4, 9]
B: [6, 7, 8, 6, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 7 6 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, 10 y2, -5 y1 - 15 y2 + 11 y5 - 5 y3 + 11 y4, 0, 0, 5 y1, 5 y2, 5 y5, 0, 5 y3, 5 y4]

  p = s 2 - s 4 - s 5 + s 7   p = - s 2 - s 3 + s 5 + s 6

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y5 + y6, y1, 0, 0, y5 - y6 + y3, y3, y4, y5, 0, y2, y5, y6]

  p = s 4 - s 7   p' = s 4 - s 7   p' = s 5 - s 8


 
1914 . Coloring, {2, 4, 5, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, B, B, 2, 4, 9]
B: [6, 7, 8, 6, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y5, y5, y4, 0, 0, y3, y2, y5, 0, y1, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y1, y2, y3, y3, 0, y4, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1915 . Coloring, {2, 4, 5, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, B, C, C, 2, 4, 9]
B: [6, 7, 8, 6, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 7 y1 + 7 y2 - 9 y3 - 9 y4 + 7 y5 - 9 y6, 7 y1 + 7 y2 - 9 y3 - 9 y4 + 7 y5 - 9 y6, 2 y1, 0, 0, 2 y2, 2 y3, 2 y4, 0, 2 y5, 2 y6]

  p = - s 3 + s 5 + s 6 - s 8   p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y6, y4, y5, y5, 0, y3, y2, y1]

  p = s 5 - s 8


 
1916 . Coloring, {2, 4, 5, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, 2, 4, 5]
B: [6, 7, 8, 6, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y2 - y3 - y4 + y5 + y6 - y7 + y8, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 8, 9, 11, 12}}    order:   7

See Matrix
 

[y1, 0, y2, 0, 0, y3, y6, y4, y5, y6, y7, y8]

  p = - s 2 + s 9


 
1917 . Coloring, {2, 4, 5, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, 2, 1, 9]
B: [6, 7, 8, 6, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y7, 3 y6, 3 y5, 0, 0, 0, 3 y4, 3 y3, 3 y2, -3 y7 - 3 y5 - 3 y3 + 8 y2 + 5 y1, -3 y6 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[0, 0, y5, y4, y3, y2, y1, y9, 0, y8, y7, y6]


 
1918 . Coloring, {2, 4, 5, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, C, 4, 9]
B: [6, 7, 8, 6, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y6, y1, 0, 0, y2, y6, y7, y5, y3, y4]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y2, y1, y10, 0, y9, y7, y8, y6, 0, y5, y4, y3]


 
1919 . Coloring, {2, 4, 5, 7, 8, 10, 11, 12}

R: [7, 8, 7, 7, 3, A, A, B, B, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y6, 0, 0, y3, y5, 2 y2, y4, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y2, 0, y1, 0, y6, y5, y4, y3, 0, y9, y7, y8]


 
1920 . Coloring, {2, 4, 5, 7, 9, 10, 11, 12}

R: [7, 8, 7, 7, 3, A, A, C, C, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y6, y4, 2 y4, 0, 0, y5, y3, y1, y2, 0, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 8, 11}, {5, 10, 12}}   

See Matrix
 

[-5 y4 + 11 y1 - 5 y2 - 5 y3 - 5 y5 + 11 y6 - 5 y7 + 11 y8, 0, 5 y4, 0, 5 y1, 5 y2, 5 y3, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1921 . Coloring, {2, 4, 5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, A, B, B, C, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y6, 3 y7, 3 y5, 0, 0, 3 y4, 3 y2, 3 y3, 3 y7, -3 y6 - 3 y5 - 3 y4 - 3 y2 + 13 y3 - 6 y7 + 13 y1, 3 y1]

  p' = s 4 + s 5 - s 7 - s 8   p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y4, y3, y7, y5, 0, y8, y7, y6]

  p = s 6 - s 9


 
1922 . Coloring, {2, 4, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, 2, 4, 5]
B: [6, 7, 8, 6, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, 2 y2 + y7 - y6 - y5 + y4 + y3 - y1, y2, y7, y6, 0, y5, y4, 0, y3, y1, y2]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8   p = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, 0, y5, y2, y3, y4, y8, y7, y6]

  p = s 3 - s 9


 
1923 . Coloring, {2, 4, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, 2, 1, 9]
B: [6, 7, 8, 6, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y1 - 3 y7 + 8 y3 - 3 y4 + 5 y6, -3 y2 + 5 y3 - 3 y5 + 8 y6, 3 y1, 0, 0, 0, 3 y2, 3 y7, 3 y3, 3 y4, 3 y5, 3 y6]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, 0, y5, y1, y2, y3, y4, y6, 0, y7, y8, y9]


 
1924 . Coloring, {2, 4, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, C, 4, 9]
B: [6, 7, 8, 6, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, y3, y2, 0, 0, y1, y3, y6, y7, y4, y5]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}, {1, 2, 6, 7, 10, 11}}    order:   12

See Matrix
 

[33 y2 - 2 y4 + 33 y6 - 2 y5 - 13 y1 - 13 y3 - 13 y7, 2 y1, -5 y1 + 13 y2 - 5 y3 + 13 y6 - 5 y7 - 2 y8, 0, 2 y2, 2 y3, 2 y4, 2 y6, 0, 2 y5, 2 y7, 2 y8]

  p = - s - s 3 + s 7 + s 9   p' = - s - s 3 + s 7 + s 9


 
1925 . Coloring, {2, 4, 6, 7, 8, 10, 11, 12}

R: [7, 8, 7, 7, A, 3, A, B, B, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y6, y7, y1, 0, 0, y2, y3, 2 y7, y4, y5, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y2, 0, y3, 0, y1, y5, y6, y7, 0, y4, y8, y9]


 
1926 . Coloring, {2, 4, 6, 7, 9, 10, 11, 12}

R: [7, 8, 7, 7, A, 3, A, C, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y2, y1, 2 y1, 0, 0, y7, y6, y5, y4, 0, y3]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 8, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y6, 0, y5, y2, y3, y4, 0, y7, y8, y9]


 
1927 . Coloring, {2, 4, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, B, B, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 5 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -6 y5 - 3 y1 - 3 y2 - 3 y3 + 13 y4 - 3 y6 + 13 y7, 3 y5, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = s 4 + s 5 - s 7 - s 8   p' = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[-y3 + y4 + y5, 0, y2, 0, y3, -y2 + y4, y5, y4 + y5 - y1, 0, y1, y5, y4]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8   p = - s 5 + s 9


 
1928 . Coloring, {2, 4, 7, 8, 9, 10, 11, 12}

R: [7, 8, 7, 7, A, A, A, B, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y2, 0, 3 y1, 0, 0, -3 y2 + 5 y6 - 3 y4 + 8 y3, -3 y1 + 8 y6 - 3 y5 + 5 y3, 3 y6, 3 y5, 3 y4, 3 y3]

  p' = - s + s 7   p = - s + s 7

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[-y1 + y2 + y5 - y3 + y4 + y6 - y7, 0, y1, 0, y2, y5, y3, y4, 0, 0, y6, y7]

  p = - s 5 + s 6 - s 7 + s 8


 
1929 . Coloring, {2, 5, 6, 7, 8, 9, 10, 11}

R: [7, 8, 7, 6, 3, 3, A, B, C, 2, 4, 5]
B: [6, 7, 8, 7, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y9, y9 - y7 + y8 + y6 + y4 - y5 - y2 + y3 - y1, y7, y8, y6, y4, y5, 0, y2, y3, y1]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9 - s 10

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y6, 2 y5, y5, y4, y3, y2, y1]

  p = - s 2 + s 8


 
1930 . Coloring, {2, 5, 6, 7, 8, 9, 10, 12}

R: [7, 8, 7, 6, 3, 3, A, B, C, 2, 1, 9]
B: [6, 7, 8, 7, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 4 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y1 - 3 y4 + 8 y5 - 3 y6 + 5 y8, -3 y2 - 3 y3 + 5 y5 - 3 y7 + 8 y8, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 3 + s 9   p' = - s 3 + s 9

Omega Rank for B :  cycles: {{4, 7, 11}, {5, 10, 12}}    order:   3

See Matrix
 

[0, 0, 0, 3 y4, 3 y3, 3 y2, 3 y1, 3 y2, 0, 3 y3, -3 y4 + 7 y3 + 7 y2 - 3 y1, 3 y3 + 3 y2]

  p' = - s 4 + s 7   p' = - s 3 + s 6   p' = - s 2 + s 5   p = s 2 - s 5


 
1931 . Coloring, {2, 5, 6, 7, 8, 9, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, B, C, C, 4, 9]
B: [6, 7, 8, 7, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, y3, y4, y5, y6, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y5, y3, y4, y6, 0, y7, y8, y9]


 
1932 . Coloring, {2, 5, 6, 7, 8, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, B, B, 2, 4, 9]
B: [6, 7, 8, 7, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, y5, y6, y4, 0, y7, y9, y8, y3, y1, y2, 0]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y2, 2 y3, y3, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1933 . Coloring, {2, 5, 6, 7, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, C, C, 2, 4, 9]
B: [6, 7, 8, 7, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, y2, y3, y1, 0, y4, y5, y6, y8, y7, 0, y9]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y5, y3, 2 y4, y4, 0, y2, y6, y7]

  p = - s 5 + s 8


 
1934 . Coloring, {2, 5, 6, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, B, B, C, 2, 4, 9]
B: [6, 7, 8, 7, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 7, 11}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y6 - 3 y3 - 3 y4 + 13 y5 - 3 y7 + 13 y8, 3 y1, 3 y2, 0, 3 y6, 3 y3, 3 y4, 3 y5, 0, 3 y7, 3 y8]

  p = s 3 + s 4 - s 8 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y4, y3, 2 y6, y6, 0, y2, y6, y5]

  p' = s 4 - s 7   p = s 4 - s 7


 
1935 . Coloring, {2, 5, 7, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, A, A, B, C, 2, 4, 9]
B: [6, 7, 8, 7, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y6 - 3 y4 + 5 y3 - 3 y2 + 8 y8, -3 y7 - 3 y5 + 8 y3 - 3 y1 + 5 y8, 3 y7, 0, 3 y6, 3 y4, 3 y5, 3 y3, 3 y1, 3 y2, 3 y8]

  p' = - s 3 + s 9   p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y3, 0, y2, 0, y1, y8, y9, y7, 0, y6, y5, y4]


 
1936 . Coloring, {2, 6, 7, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, A, 3, A, B, C, 2, 4, 9]
B: [6, 7, 8, 7, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 3, 4, 6, 7, 8, 10, 11}, {9, 12}}    order:   8

See Matrix
 

[0, 3 y7, 3 y8, 3 y6, 0, 3 y5, 3 y4, 3 y2, 3 y3, -3 y8 - 3 y6 - 3 y2 + 8 y3 + 5 y1, -3 y7 - 3 y5 - 3 y4 + 5 y3 + 8 y1, 3 y1]

  p' = - s + s 9   p = - s + s 9

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, y3, y2, y1, -y5 + y4 - y3 + y2 - y1 - y8 + y7 + y6, 0, y8, y7, y6]

  p = - s 6 + s 7 - s 8 + s 9


 
1937 . Coloring, {3, 4, 5, 6, 7, 8, 9, 10}

R: [7, 7, 8, 7, 3, 3, A, B, C, 2, 1, 5]
B: [6, 8, 7, 6, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[y2, y1, y3, 0, y4, 0, y5, y6, 0, y8, y7, y9]

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - 2 y3 - y2 - y4 + y5 + y6, 0, y1, y3, y3, y2, y4, y5, y6]

  p' = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = s 2 - s 8


 
1938 . Coloring, {3, 4, 5, 6, 7, 8, 9, 11}

R: [7, 7, 8, 7, 3, 3, A, B, C, C, 4, 5]
B: [6, 8, 7, 6, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 8 vs 9

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, y3, 0, y5, y4, 0, y6, y8, y7]

Omega Rank for B :  cycles: {{1, 2, 6, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[y1 + y3 - y2 - y4 - y5 - y6 + y7 + y8, y1, 0, 0, 0, y3, y2, y4, y5, y6, y7, y8]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9


 
1939 . Coloring, {3, 4, 5, 6, 7, 8, 9, 12}

R: [7, 7, 8, 7, 3, 3, A, B, C, C, 1, 9]
B: [6, 8, 7, 6, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y4, 0, y3, 0, 0, 0, y2, y1, y8, y7, y6, y5]

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[0, y9, 0, y8, y7, y6, y5, y4, 0, y3, y2, y1]


 
1940 . Coloring, {3, 4, 5, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, 2, 4, 5]
B: [6, 8, 7, 6, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y3, y4, 0, y5, y6, 0, y7, y8, 0]

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y1 + 2 y6 + y5 + y4 - y3 - y2, y6, y6, y5, y4, y3, y2]

  p' = s 6 - s 7   p = s 6 - s 8


 
1941 . Coloring, {3, 4, 5, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, 2, 1, 9]
B: [6, 8, 7, 6, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y2, y1, y5, 0, 0, 0, y7, y6, y5, y4, y3, 0]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y3, y2, y1, y1, 0, y7, y6, y5]

  p = - s 5 + s 8


 
1942 . Coloring, {3, 4, 5, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, C, 4, 9]
B: [6, 8, 7, 6, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1, y2, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2, y1, 0, 0, y3, y4, y5, y8, 0, y6, y7, y9]


 
1943 . Coloring, {3, 4, 5, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, 2, 4, 5]
B: [6, 8, 7, 6, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 5, 8, 12}}    order:   12

See Matrix
 

[0, 7 y7, 7 y5, 7 y6, 7 y4, 0, 7 y3, 7 y2, 0, -7 y7 + 9 y5 - 7 y6 + 9 y4 - 7 y3 + 9 y2 + 9 y1, 0, 7 y1]

  p = - s 2 - s 3 - s 4 + s 6 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, 0, y5, y4, y4, y3, y2, y1, y6 - y5 + 2 y4 + y3 + y2 - y1]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1944 . Coloring, {3, 4, 5, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, 2, 1, 9]
B: [6, 8, 7, 6, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[7 y6 + 7 y5 - 9 y4 - 9 y3 + 7 y2 - 9 y1, 2 y6, 7 y6 + 7 y5 - 9 y4 - 9 y3 + 7 y2 - 9 y1, 0, 0, 0, 2 y5, 2 y4, 2 y3, 2 y2, 0, 2 y1]

  p = s 3 - s 5 - s 6 + s 8   p' = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1, y3, y4, y2, y2, 0, y7, y5, y6]

  p = - s 5 + s 8


 
1945 . Coloring, {3, 4, 5, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, C, 4, 9]
B: [6, 8, 7, 6, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 8 vs 9 5 vs 7 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y1 - y2, y1 - y2, 0, 0, y1, y2, y3, y4, 0, y5]

  p' = - s 4 + s 6   p = - s 4 + s 6

Omega Rank for B :  cycles: {{1, 2, 6, 8, 10, 11}}    order:   6

See Matrix
 

[y2, y1, 0, 0, 2 y5, y6, y5, y3, 0, y4, y7, 0]

  p = - s 2 + s 8


 
1946 . Coloring, {3, 4, 5, 6, 7, 10, 11, 12}

R: [7, 7, 8, 7, 3, 3, A, C, B, 2, 4, 9]
B: [6, 8, 7, 6, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y2, y1, y9, 0, 0, y8, y6, y5, y7, y4, y3]

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y4, y5, y6, y6, 0, y3, y1, y2]

  p = s 5 - s 8


 
1947 . Coloring, {3, 4, 5, 6, 8, 9, 10, 11}

R: [7, 7, 8, 7, 3, 3, B, B, C, 2, 4, 5]
B: [6, 8, 7, 6, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y1, y7, y6, y5, 0, y4, y3, 0, 0, y2, y1]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, 0, y2, y3, y3, y4, y5, y6, y1]

  p = - s 2 + s 8


 
1948 . Coloring, {3, 4, 5, 6, 8, 9, 10, 12}

R: [7, 7, 8, 7, 3, 3, B, B, C, 2, 1, 9]
B: [6, 8, 7, 6, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[3 y5, 3 y6, 3 y6, 0, 0, 0, 3 y2, 3 y3, 3 y4, 0, -3 y5 - 6 y6 - 3 y2 - 3 y3 + 13 y4 + 13 y1, 3 y1]

  p = - s 3 + s 5 + s 6 - s 8   p' = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y3, y1, y5, y5, 0, y4, y5, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
1949 . Coloring, {3, 4, 5, 6, 8, 9, 11, 12}

R: [7, 7, 8, 7, 3, 3, B, B, C, C, 4, 9]
B: [6, 8, 7, 6, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 7 8 vs 9

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 0, 5 y1, -5 y1 - 5 y4 - 5 y5 + 11 y3 - 5 y2 + 11 y6, 0, 0, 5 y4, 5 y5, 5 y3, 0, 5 y2, 5 y6]

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, y4, y7, y5, 0, y6, y7, y8]

  p = - s 4 + s 9


 
1950 . Coloring, {3, 4, 5, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, B, B, 2, 4, 9]
B: [6, 8, 7, 6, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 5 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y5, y5, y4, 0, 0, y3, y2, y5, 0, y1, 0]

  p' = s 3 - s 6   p = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[2 y3, 0, 0, 0, y4, y2, y3, y3, 0, y1, 0, y5]

  p = - s 3 + s 6   p' = - s 3 + s 6


 
1951 . Coloring, {3, 4, 5, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, B, C, C, 2, 4, 9]
B: [6, 8, 7, 6, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 7 y6 + 7 y5 - 9 y1 - 9 y2 + 7 y3 - 9 y4, 7 y6 + 7 y5 - 9 y1 - 9 y2 + 7 y3 - 9 y4, 2 y6, 0, 0, 2 y5, 2 y1, 2 y2, 0, 2 y3, 2 y4]

  p = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 5 + s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, y4, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1952 . Coloring, {3, 4, 5, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, 2, 4, 5]
B: [6, 8, 7, 6, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y9, y7, y8, y6, 0, y5, y4, 0, y3, y2, y1]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y8, 0, y7, 0, 0, y6, y4, y5, y3, y5, y2, y1]

  p = s 4 - s 9


 
1953 . Coloring, {3, 4, 5, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, 2, 1, 9]
B: [6, 8, 7, 6, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y8, 3 y6, 3 y7, 0, 0, 0, 3 y5, 3 y3, 3 y4, -3 y8 - 3 y6 - 3 y7 - 3 y5 - 3 y3 + 13 y4 - 3 y2 + 13 y1, 3 y2, 3 y1]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {3, 4, 6, 7, 11}}   

See Matrix
 

[0, 0, 7 y5, 7 y4, 7 y2, 7 y3, -7 y5 - 7 y4 + 9 y2 - 7 y3 + 9 y1 + 9 y8 - 7 y6 + 9 y7, 7 y1, 0, 7 y8, 7 y6, 7 y7]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1954 . Coloring, {3, 4, 5, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, C, 4, 9]
B: [6, 8, 7, 6, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 5 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y7, y6, 0, 0, y4, y5, y3, y1, y2, -y7 + y6 - y4 + y5 + y3 + y1 - y2]

  p = - s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, 5 y1 + 3 y3 - 8 y2 - y5, 24 y1 + 19 y3 - 37 y2 - 7 y5 - 8 y4, 0, 9 y1 + 7 y3 - 14 y2 - 2 y5 - 3 y4, 11 y1 + 7 y3 - 17 y2 - 3 y5 - y4, y3, 14 y1 + 11 y3 - 21 y2 - 4 y5 - 5 y4, 0, y2, y5, y4]

  p' = s 2 - s 7   p' = s 4 - s 9   p' = s 3 - s 8   p = s - s 6   p' = s - s 6


 
1955 . Coloring, {3, 4, 5, 7, 8, 10, 11, 12}

R: [7, 7, 8, 7, 3, A, A, B, B, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y7, y6, 0, 0, y5, y4, 2 y7, y3, y2, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[y4, 0, y3, 0, y1, y2, -y4 - y3 + y1 - y2 + y8 + y7 - y5 + y6, y8, 0, y7, y5, y6]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1956 . Coloring, {3, 4, 5, 7, 9, 10, 11, 12}

R: [7, 7, 8, 7, 3, A, A, C, C, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 3 y1, 3 y5 + 3 y4 - 3 y2, 6 y5 + 6 y4 - 6 y2, 0, 0, -3 y1 + 4 y5 + 4 y4 + 6 y2 - 3 y3, 3 y5, 3 y4, 3 y3, 0, 3 y2]

  p' = s 4 - s 7   p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[-5 y1 + 11 y2 - 5 y3 - 5 y4 - 5 y5 + 11 y6 - 5 y7 + 11 y8, 0, 5 y1, 0, 5 y2, 5 y3, 5 y4, 5 y5, 0, 5 y6, 5 y7, 5 y8]

  p = - s 2 - s 3 - s 4 + s 7 + s 8 + s 9


 
1957 . Coloring, {3, 4, 5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, A, B, B, C, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y6, 3 y5, 3 y4, 0, 0, 3 y3, 3 y6 - 3 y5, 3 y2, 3 y5, -6 y6 - 3 y5 - 3 y4 - 3 y3 + 13 y2 + 13 y1, 3 y1]

  p' = - s 4 - s 5 + s 7 + s 8   p' = s 3 + s 4 - s 6 - s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, y3, 0, y1, y5, y6, y7, 0, y4, y7, y8]

  p = s 6 - s 9


 
1958 . Coloring, {3, 4, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, 2, 4, 5]
B: [6, 8, 7, 6, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y5 - y7, y4, y5, 0, y6, y7, 0, y2, y3, y5 - y7]

  p = - s 5 + s 8   p' = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, y6, 0, 0, y7, y8, y6, y5, y4, y3, y2]

  p = - s 3 + s 9


 
1959 . Coloring, {3, 4, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, 2, 1, 9]
B: [6, 8, 7, 6, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y1, -3 y1 - 3 y2 - 3 y6 - 3 y7 + 13 y8 - 3 y3 - 3 y4 + 13 y5, 3 y2, 0, 0, 0, 3 y6, 3 y7, 3 y8, 3 y3, 3 y4, 3 y5]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y6, y5, y2, y3, y4, y1, 0, y7, y8, y9]


 
1960 . Coloring, {3, 4, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, C, 4, 9]
B: [6, 8, 7, 6, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 8 10 vs 10

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y4, y3, 0, 0, y2, y1, y4 - y3 + y2 - y1 - y7 + y6 + y5, y7, y6, y5]

  p = s 7 - s 8

Omega Rank for B :  cycles: {{1, 2, 3, 5, 6, 7, 8, 10, 11, 12}}    order:   10

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, y7, 0, y8, y9, y10]


 
1961 . Coloring, {3, 4, 6, 7, 8, 10, 11, 12}

R: [7, 7, 8, 7, A, 3, A, B, B, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y5, y6, y4, 0, 0, y3, y2, 2 y6, y1, y7, 0]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y5, 0, y4, 0, y3, y2, y1, y9, 0, y8, y7, y6]


 
1962 . Coloring, {3, 4, 6, 7, 9, 10, 11, 12}

R: [7, 7, 8, 7, A, 3, A, C, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 5 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y4 + 4 y1 + 4 y2 - 3 y3 + 6 y5, 3 y1 + 3 y2 - 3 y5, 6 y1 + 6 y2 - 6 y5, 0, 0, 3 y4, 3 y1, 3 y2, 3 y3, 0, 3 y5]

  p' = s 3 - s 6   p' = s 4 - s 7   p = s 3 - s 6

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, y8, y5, 0, y6, y7, y9]


 
1963 . Coloring, {3, 4, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, B, B, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 6 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y2 + 3 y6, 3 y2, 3 y1, 0, 0, -9 y2 - 6 y6 - 3 y1 + 13 y5 - 3 y3 + 13 y4, 3 y6, 3 y5, 3 y2, 3 y3, 3 y4]

  p' = s 4 + s 5 - s 7 - s 8   p = s 3 - s 9   p' = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y1, 0, y5, 0, y2, y3, y4, y7, 0, y6, y7, y8]

  p = - s 4 + s 9


 
1964 . Coloring, {3, 4, 7, 8, 9, 10, 11, 12}

R: [7, 7, 8, 7, A, A, A, B, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y7 - 3 y6 + 13 y5 - 3 y4 - 3 y3 + 13 y2, 0, 3 y1, 0, 0, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3, 3 y2]

  p = - s 4 - s 5 + s 7 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y5, 0, y4, y2, y3, y6, 0, 0, y8, y7]


 
1965 . Coloring, {3, 5, 6, 7, 8, 9, 10, 11}

R: [7, 7, 8, 6, 3, 3, A, B, C, 2, 4, 5]
B: [6, 8, 7, 7, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, y5, -y1 - y2 - y3 + 5 y5 - y4 - y6 - y7, y1, y2, y3, y5, y4, 0, y5, y6, y7]

  p = s 3 - s 8   p' = - s 4 + s 9   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y5, 2 y2, y2, y3, y4, y6, y7]

  p = - s 2 + s 8


 
1966 . Coloring, {3, 5, 6, 7, 8, 9, 10, 12}

R: [7, 7, 8, 6, 3, 3, A, B, C, 2, 1, 9]
B: [6, 8, 7, 7, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 4 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y1, 3 y2, -3 y1 - 3 y2 - 3 y9 - 3 y7 - 3 y8 + 13 y6 - 3 y5 - 3 y4 + 13 y3, 0, 0, 3 y9, 3 y7, 3 y8, 3 y6, 3 y5, 3 y4, 3 y3]

  p = s 6 + s 7 - s 9 - s 10

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, -3 y1 - 3 y4 + 7 y3, 3 y2, -3 y2 + 3 y3, 3 y1, -3 y2 + 3 y3, 0, 3 y2, 3 y4, 3 y3]

  p = s 2 - s 8   p' = s 3 - s 6   p' = s 4 - s 7   p' = s 2 - s 5


 
1967 . Coloring, {3, 5, 6, 7, 8, 9, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, B, C, C, 4, 9]
B: [6, 8, 7, 7, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 0, 7 y2, 7 y1, 0, 7 y6, 7 y5, 7 y4, 9 y2 + 9 y1 + 9 y6 - 7 y5 + 9 y4 - 7 y3 + 9 y8 - 7 y7, 7 y3, 7 y8, 7 y7]

  p = - s 3 - s 4 + s 8 + s 9

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y1, y2, 0, 0, y3, y4, y5, y6, 0, y7, y8, y9]


 
1968 . Coloring, {3, 5, 6, 7, 8, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, B, B, 2, 4, 9]
B: [6, 8, 7, 7, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {3, 4, 6, 8, 11}}   

See Matrix
 

[0, y5, 5 y5 - y1 - y2 - y3 - y4 - y6, y1, 0, y2, y5, y3, y4, y5, y6, 0]

  p = - s 2 + s 7   p' = - s 2 + s 7   p' = - s 3 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y3, y2, 2 y4, y4, 0, y5, y6, y7]

  p = - s 5 + s 8


 
1969 . Coloring, {3, 5, 6, 7, 9, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, C, C, 2, 4, 9]
B: [6, 8, 7, 7, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 9 vs 9 9 vs 9 6 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, y6, y5, y4, 0, y3, y6, y2, y1, y6, 0, -y5 - y4 - y3 - y2 - y1 + 5 y6]

  p' = s 6 - s 8   p' = s 5 - s 7   p = s 5 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, y1, y7, 2 y6, y6, 0, y5, y4, y3]

  p = s 5 - s 8


 
1970 . Coloring, {3, 5, 6, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, B, B, C, 2, 4, 9]
B: [6, 8, 7, 7, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 3 y8, 3 y7, 3 y6, 0, 3 y5, 3 y4, 3 y3, 3 y2, 0, -3 y8 - 3 y7 - 3 y6 - 3 y5 - 3 y4 - 3 y3 + 13 y2 + 13 y1, 3 y1]

  p = - s 3 - s 4 + s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 2 y5, y5, 0, y4, y5, y6]

  p = s 4 - s 7   p' = - s 4 + s 7


 
1971 . Coloring, {3, 5, 7, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, 3, A, A, B, C, 2, 4, 9]
B: [6, 8, 7, 7, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 8 vs 9

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y2, 3 y3, 3 y4, 0, -3 y2 - 3 y3 - 3 y4 - 3 y1 - 3 y8 + 13 y9 - 3 y5 - 3 y6 + 13 y7, 3 y1, 3 y8, 3 y9, 3 y5, 3 y6, 3 y7]

  p = - s 6 - s 7 + s 9 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[-7 y1 + 9 y2 - 7 y3 - 7 y4 + 9 y5 + 9 y6 - 7 y7 + 9 y8, 0, 7 y1, 0, 7 y2, 7 y3, 7 y4, 7 y5, 0, 7 y6, 7 y7, 7 y8]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
1972 . Coloring, {3, 6, 7, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, A, 3, A, B, C, 2, 4, 9]
B: [6, 8, 7, 7, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 4 vs 9

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {2, 7, 10}, {9, 12}}   

See Matrix
 

[0, -3 y3 + 7 y5 - 3 y8 + 7 y7, -3 y1 - 3 y2 - 3 y4 + 6 y5 - 3 y6 + 6 y7, 3 y1, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y8, 3 y6, 3 y7]

  p = - 2s - 3s 2 - 2s 3 + s 5 + 2s 6 + 3s 7 + 2s 8 - s 10   p = s + 2s 2 + 2s 3 + s 4 - s 6 - 2s 7 - 2s 8 - s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y1 + y4 + y3 - y2, 0, y1, y1 + y4 + y3 - y2, y4 + y3, y4, 0, y3, y2, y2]

  p = - s 2 + s 6   p = - s 2 + s 5 + s 7 - s 8   p' = s 3 - s 7   p' = s 2 - s 6   p = - s 2 + s 7 - s 8 + s 9


 
1973 . Coloring, {4, 5, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, 2, 4, 5]
B: [6, 8, 8, 6, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y2, y4, y4, 0, y3, 0, 0, y5, y6, y6]

  p = - s 4 + s 7   p' = - s 4 + s 7

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 - y6 - y5 - y4 + y3 + y2, 0, 0, 0, 0, y1, 0, y6, y5, y4, y3, y2]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
1974 . Coloring, {4, 5, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, 2, 1, 9]
B: [6, 8, 8, 6, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[3 y4, 3 y5, 6 y1, 0, 0, 0, 3 y3, 0, 3 y2, -3 y4 - 3 y5 - 3 y3 + 13 y2 - 9 y1 + 13 y6, 3 y1, 3 y6]

  p = s 3 + s 4 - s 6 - s 7   p' = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, 2 y5, 2 y4, 2 y3, 0, 2 y2, 0, 2 y1, 3 y2, 2 y6]

  p = s 4 - s 7


 
1975 . Coloring, {4, 5, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, C, 4, 9]
B: [6, 8, 8, 6, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 2 y3, y1, 0, 0, y1 + y5 + y4 + y3 - y2, 0, y5, y4, y3, y2]

  p' = s 5 - s 6   p = s 5 - s 7

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2, y1, 0, 0, y4, y3, 0, y5, 0, y8, y7, y6]


 
1976 . Coloring, {4, 5, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 + 24s 5 + 16s 6 + 96s 7 + 64s 8   p = - 3s 2 + 2s 3 + 8s 5 + 32s 7

R: [7, 7, 7, 7, 3, 3, A, B, B, 2, 4, 9]
B: [6, 8, 8, 6, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 7 6 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y1, y6, y2, 0, 0, y5, 0, y6, y3, y4, 0]

  p = - s 4 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, y5, y4, 0, y3, 0, y2, y3, y1]

  p = s 4 - s 7


 
1977 . Coloring, {4, 5, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 5 + 32s 7   p = s 2 - 4s 4 - 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 7, 7, 7, 3, 3, A, C, C, 2, 4, 9]
B: [6, 8, 8, 6, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, -2 y1 - y2 + 6 y4 - y3, y1, y1, 0, 0, y2, 0, y4, y3, 0, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = - s 3 + s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y6, y5, 0, y4, 0, y3, y2, y1]


 
1978 . Coloring, {4, 5, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, B, B, C, 2, 4, 9]
B: [6, 8, 8, 6, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 7 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y4, 3 y4, 3 y5, 0, 0, 3 y3, 0, 3 y1, 0, -6 y4 - 3 y5 - 3 y3 + 13 y1 + 13 y2, 3 y2]

  p = - s 2 - s 3 + s 5 + s 6   p = s 2 - s 4 - s 5 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 0, 2 y5, 0, y4, y5, y6]

  p = - s 4 + s 7


 
1979 . Coloring, {4, 5, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 5 - 32s 7   p = s 2 - 4s 4 + 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 7, 7, 7, 3, A, A, B, C, 2, 4, 9]
B: [6, 8, 8, 6, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y2 - 3 y1 + 13 y6 - 3 y5 - 6 y4 + 13 y3, 3 y4, 3 y2, 0, 0, 3 y1, 0, 3 y6, 3 y5, 3 y4, 3 y3]

  p' = s 3 + s 4 - s 6 - s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y5, 0, y4, y3, 0, y2, 0, y8, y7, y6]


 
1980 . Coloring, {4, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 9s 2 - 28s 4 - 40s 5 - 16s 6 + 96s 7 + 64s 8   p' = 3s 2 - 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 7, 7, 7, A, 3, A, B, C, 2, 4, 9]
B: [6, 8, 8, 6, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 3 y1, 3 y6, -3 y1 - 6 y6 - 3 y2 + 13 y4 - 3 y5 + 13 y3, 0, 0, 3 y2, 0, 3 y4, 3 y5, 3 y6, 3 y3]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y3, y4, 0, y5, 0, y6, y7, y8]


 
1981 . Coloring, {5, 6, 7, 8, 9, 10, 11, 12}

R: [7, 7, 7, 6, 3, 3, A, B, C, 2, 4, 9]
B: [6, 8, 8, 7, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, 3 y8, 3 y7, 3 y6, 0, 3 y5, 3 y4, 0, 3 y3, 3 y2, -3 y8 - 3 y7 - 3 y6 - 3 y5 - 3 y4 + 13 y3 - 3 y2 + 13 y1, 3 y1]

  p = s 5 + s 6 - s 8 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, 2 y4, 0, y5, y6, y7]

  p = s 5 - s 8


 
1982 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 10}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, 2, 1, 5]
B: [6, 7, 7, 6, A, A, B, C, B, C, 4, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y5, y4, y3, 0, y2, 0, y1, -y5 + y4 + y3 - y2 + y1 - y8 + y7 + y6, 0, y8, y7, y6]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{4, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, 0, y1 - y2 - y3 - y4 + y5 + y6, 0, y1, y2, 0, y3, y4, y5, y6]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1983 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 11}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, C, 4, 5]
B: [6, 7, 7, 6, A, A, B, C, B, 2, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 5 vs 8 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 5, 7, 8, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4 + y5 - y2, -y1 + y4 + y5, y1, 0, y2, -y3 + y4 + y5, 0, y3, y4, y5]

  p = - s + s 2 - s 5 + s 6   p = - s + s 3 - s 5 + s 7   p = - s + s 4 - s 5 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y1 - y2 + y3 + y4 + y5 - y7 - y6, 0, 0, 0, y2, y3, 0, y4, y5, y7, y6]

  p = s 3 - s 4 + s 5 - s 6 + s 7 - s 8


 
1984 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, C, 1, 9]
B: [6, 7, 7, 6, A, A, B, C, B, 2, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[y1, 0, y1 - y7 + y6 + y5 + y3 - y4 - y2, 0, 0, 0, y7, y6, y5, y3, y4, y2]

  p = - s 7 + s 8

Omega Rank for B :  cycles: {{2, 4, 6, 7, 10, 11}}    order:   6

See Matrix
 

[0, y1, 0, y2, y5, y3, y4, 0, 0, y6, y7, y8]


 
1985 . Coloring, {2, 3, 4, 5, 6, 7, 8, 10, 11}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 - 8s 5 + 32s 7   p = - 9s 2 + 4s 4 - 24s 5 - 16s 6 + 96s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, A, B, B, 2, 4, 5]
B: [6, 7, 7, 6, A, A, B, C, C, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y7, y6, y5, y4, 0, y3, y2, 0, y1, -y7 - y6 + y5 + y4 - y3 + y2 + y1, 0]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1 - y2 - y3 - y4 + y5 + y6, 0, 0, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p = - s 6 + s 7


 
1986 . Coloring, {2, 3, 4, 5, 6, 7, 8, 10, 12}

Ωp(Δ)=0:     p = 3s 2 + 10s 3 + 40s 5 + 32s 6 + 32s 7 + 128s 8   p' = - 3s 2 + 2s 3 - 8s 4 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, 3, A, B, B, 2, 1, 9]
B: [6, 7, 7, 6, A, A, B, C, C, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 - y3 - y5 + y4, y1, y6, 0, 0, 0, y2, y3, y6, y5, y4, 0]

  p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y3, y1, y2, y7, 0, 0, y6, y4, y5]


 
1987 . Coloring, {2, 3, 4, 5, 6, 7, 8, 11, 12}

Ωp(Δ)=0:     p' = 3s 2 + 2s 3 + 8s 4 - 8s 5 - 32s 7   p = 3s 2 - 10s 3 - 40s 5 + 32s 6 - 32s 7 + 128s 8

R: [7, 8, 8, 7, 3, 3, A, B, B, C, 4, 9]
B: [6, 7, 7, 6, A, A, B, C, C, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{4, 7, 9, 10, 11, 12}}    order:   6

See Matrix
 

[0, 0, y1 - y2 + y3 + y4 + y7 - y5 - y6, y1, 0, 0, y2, y3, y4, y7, y5, y6]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y4 + y2 - y3 - y6 + y7 - y5, y1, 0, 0, y4, y2, y3, 0, 0, y6, y7, y5]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
1988 . Coloring, {2, 3, 4, 5, 6, 7, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 + 8s 5 + 16s 6 + 32s 7 + 64s 8   p' = s 2 - 2s 3 + 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, 3, A, C, C, 2, 4, 5]
B: [6, 7, 7, 6, A, A, B, B, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 5 vs 8 6 vs 7

Omega Rank for R :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[0, y4, -y4 + y3 + y2, -y1 + y3 + y2, y3, 0, -y5 + y3 + y2, y1, 0, y2, 0, y5]

  p = - s 5 + s 6   p = - s 5 + s 7   p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6, 0, 0, 0, 0, y3, y4, 0, y5, y2, y6 - y3 + y4 + y5 + y2 - y1, y1]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1989 . Coloring, {2, 3, 4, 5, 6, 7, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, 3, A, C, C, 2, 1, 9]
B: [6, 7, 7, 6, A, A, B, B, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[y1, -y2 + y3 + y4 + y5 - y6, y1, 0, 0, 0, y2, y3, y4, y5, 0, y6]

  p' = s 6 - s 7   p = s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y7, y6, y5, y4, 0, 0, y3, y2, y1]


 
1990 . Coloring, {2, 3, 4, 5, 6, 7, 9, 11, 12}

Ωp(Δ)=0:     p = s 2 - 12s 4 + 8s 5 + 16s 6 + 32s 7 - 64s 8   p' = s 2 + 2s 3 - 8s 4 - 8s 5 + 32s 7

R: [7, 8, 8, 7, 3, 3, A, C, C, C, 4, 9]
B: [6, 7, 7, 6, A, A, B, B, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 4 vs 7 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   4

See Matrix
 

[0, 0, y4, y4, 0, 0, y3, y3, y2, y1, 0, y2 + y1]

  p' = s 4 - s 6   p' = s 5 - s 6   p = s 4 - s 7

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y7, y6, y5, 0, 0, y4, y3, 0]


 
1991 . Coloring, {2, 3, 4, 5, 6, 7, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, C, B, 2, 4, 9]
B: [6, 7, 7, 6, A, A, B, B, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 9, 10, 11, 12}}    order:   8

See Matrix
 

[0, -y1 + y2 - y3 + y7 + y8 + y6 - y4 - y5, y1, y2, 0, 0, y3, y7, y8, y6, y4, y5]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y5, y6, y4, 0, 0, y3, y1, y2]


 
1992 . Coloring, {2, 3, 4, 5, 6, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 + 8s 5 + 8s 6 + 16s 7   p' = s 3 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, C, 2, 4, 5]
B: [6, 7, 7, 6, A, A, A, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 8 6 vs 7

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   6

See Matrix
 

[0, 2 y7, y1, y5, y6, 0, y4, y3, 0, 0, y2, y7]

  p = - s 5 + s 8

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, 0, y1 - y2 + y3 + y4 - y5 - y6, y2, 0, y3, y4, y5, y6]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
1993 . Coloring, {2, 3, 4, 5, 6, 8, 9, 10, 12}

Ωp(Δ)=0:     p' = s 3 - 8s 6 - 16s 7   p = s 3 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, C, 2, 1, 9]
B: [6, 7, 7, 6, A, A, A, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{1, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[-6 y1 - 3 y3 - 3 y4 + 13 y2 - 3 y6 + 13 y5, 3 y1, 3 y1, 0, 0, 0, 3 y3, 3 y4, 3 y2, 0, 3 y6, 3 y5]

  p' = - s 3 - s 4 + s 6 + s 7   p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y5, y4, y3, 2 y1, 0, 0, y2, y1, y6]

  p = - s 4 + s 7


 
1994 . Coloring, {2, 3, 4, 5, 6, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, B, B, C, C, 4, 9]
B: [6, 7, 7, 6, A, A, A, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 7 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 0, -5 y1 - 5 y6 - 5 y2 + 11 y3 - 5 y4 + 11 y5, 5 y1, 0, 0, 5 y6, 5 y2, 5 y3, 0, 5 y4, 5 y5]

  p = s 3 + s 4 - s 6 - s 7

Omega Rank for B :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[y6, y5, 0, 0, y6, y4, y3, 0, 0, y2, y1, y1]

  p' = - s 4 + s 7   p = - s 4 + s 7


 
1995 . Coloring, {2, 3, 4, 5, 6, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 + 8s 5 + 48s 6 + 128s 8   p' = s 2 - 8s 4 - 24s 5 - 16s 6 - 64s 7   p' = s 3 + 4s 4 + 8s 5 + 8s 6 + 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, B, 2, 4, 9]
B: [6, 7, 7, 6, A, A, A, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 5 vs 7 5 vs 6

Omega Rank for R :  cycles: {{4, 7, 11}}    order:   3

See Matrix
 

[0, y1, y1, y4, 0, 0, y3, y2, y1, 0, y5, 0]

  p' = s 3 - s 6   p = s 3 - s 6

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   3

See Matrix
 

[y5, 0, 0, 0, y4, y3, y5, 0, 0, y2, 0, y1]

  p = - s 3 + s 6


 
1996 . Coloring, {2, 3, 4, 5, 6, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 3 + 8s 6 + 16s 7   p' = s 2 + 8s 5 + 16s 6   p = s 2 + 8s 5 + 16s 6

R: [7, 8, 8, 7, 3, 3, B, C, C, 2, 4, 9]
B: [6, 7, 7, 6, A, A, A, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 8 8 vs 8 4 vs 8 6 vs 7

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, y1, y1, y4, 0, 0, y4, y2, y3, 0, y4, -2 y1 + 5 y4 - y2 - y3]

  p' = - s 3 + s 5   p = s 3 - s 7   p' = s 4 - s 6   p' = - s 3 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y5, 0, 0, y4, y5, y6]

  p = - s 4 + s 7


 
1997 . Coloring, {2, 3, 4, 5, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 2 - 4s 4 - 8s 5 - 16s 6 - 32s 7 - 64s 8   p' = s 2 - 2s 3 - 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, A, B, C, 2, 4, 5]
B: [6, 7, 7, 6, A, 3, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 9 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y4, y3, 0, y6, y5, 0, y9, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y8, 0, y7, 0, 0, y5, y6, 0, y1, y2, y4, y3]


 
1998 . Coloring, {2, 3, 4, 5, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, A, A, B, C, 2, 1, 9]
B: [6, 7, 7, 6, A, 3, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y2 + 8 y6 - 3 y5 + 5 y7, -3 y1 - 3 y3 + 5 y6 - 3 y4 + 8 y7, 3 y1, 0, 0, 0, 3 y3, 3 y2, 3 y6, 3 y5, 3 y4, 3 y7]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 4, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[0, 0, 3 y2, 3 y3, 3 y4, -3 y2 - 3 y3 + 5 y4 - 3 y1 + 5 y6 - 3 y7 + 5 y5, 3 y1, 0, 0, 3 y6, 3 y7, 3 y5]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
1999 . Coloring, {2, 3, 4, 5, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 2 + 4s 4 - 8s 5 - 16s 6 - 32s 7 + 64s 8   p' = s 2 + 2s 3 + 8s 4 + 8s 5 - 32s 7

R: [7, 8, 8, 7, 3, A, A, B, C, C, 4, 9]
B: [6, 7, 7, 6, A, 3, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y2, 0, 0, y3, y4, y5, y6, y7, y8]

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, y2, y3, 0, y4, y5, y6, 0, 0, y7, y8, y9]


 
2000 . Coloring, {2, 3, 4, 5, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 5 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, B, B, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, 0, y4, y5, 2 y2, y6, y7, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[7 y6, 0, 7 y5, 0, 7 y4, 7 y3, 7 y2, 0, 0, 7 y1, -7 y6 - 7 y5 + 9 y4 - 7 y3 - 7 y2 + 9 y1 + 9 y7, 7 y7]

  p = s + s 2 + s 3 - s 6 - s 7 - s 8


 
2001 . Coloring, {2, 3, 4, 5, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, A, A, C, C, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y5, y6, 2 y6, 0, 0, y4, y2, y3, y1, 0, y7]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[-5 y2 + 11 y3 - 5 y1 - 5 y7 + 11 y6 - 5 y4 + 11 y5, 0, 5 y2, 0, 5 y3, 5 y1, 5 y7, 0, 0, 5 y6, 5 y4, 5 y5]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
2002 . Coloring, {2, 3, 4, 5, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 3 - 8s 6 - 16s 7   p = s 2 - 8s 5 - 16s 6   p' = s 2 - 8s 5 - 16s 6

R: [7, 8, 8, 7, 3, A, B, B, C, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -6 y5 - 3 y2 - 3 y1 - 3 y3 + 13 y4 - 3 y6 + 13 y7, 3 y5, 3 y2, 0, 0, 3 y1, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = s 4 - s 6 - s 7 + s 9   p' = s 4 + s 5 - s 7 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, y2, 0, y3, y4, y5, 0, 0, y6, y7, y8]


 
2003 . Coloring, {2, 3, 4, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = 9s 2 + 20s 4 + 40s 5 + 16s 6 + 96s 7 + 64s 8   p' = 3s 2 - 2s 3 + 8s 4 + 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, A, B, C, 2, 4, 5]
B: [6, 7, 7, 6, 3, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y7, y6, y5, y4, 0, y3, y2, 0, y1, -y7 + y5 - y4 - y3 + y2 + y1, y6]

  p' = s 3 - s 4 + s 5 - s 6 + s 7 - s 8   p = s 3 - s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y5, 0, y4, 0, 0, y3, y2, 0, y1, y8, y7, y6]


 
2004 . Coloring, {2, 3, 4, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = - 3s 2 + 2s 3 - 8s 5 + 32s 7   p = - 9s 2 + 4s 4 - 24s 5 - 16s 6 + 96s 7 + 64s 8

R: [7, 8, 8, 7, A, 3, A, B, C, 2, 1, 9]
B: [6, 7, 7, 6, 3, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {1, 2, 7, 8, 10, 11}}    order:   6

See Matrix
 

[-3 y3 + 8 y4 - 3 y5 + 5 y7, -3 y1 - 3 y2 + 5 y4 - 3 y6 + 8 y7, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p' = s 2 - s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 4, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[0, 0, y4, y1, y2, y3, y5, 0, 0, y6, y7, y8]


 
2005 . Coloring, {2, 3, 4, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 + 24s 5 - 16s 6 - 96s 7 + 64s 8   p = 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, A, B, C, C, 4, 9]
B: [6, 7, 7, 6, 3, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y4, y3, 0, 0, y2, y1, y6, y7, y8, y5]

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y6, y5, 0, y4, y2, y3, 0, 0, y9, y8, y7]


 
2006 . Coloring, {2, 3, 4, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, B, B, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y2, y3, 0, 0, y4, y5, 2 y2, y6, y7, 0]

  p = s 2 - s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y5, 0, y4, y3, y2, 0, 0, y7, y8, y6]


 
2007 . Coloring, {2, 3, 4, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 7, A, 3, A, C, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y2, 2 y2, 0, 0, y5, y4, y3, y6, 0, y7]

  p = s 6 - s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y6, y3, y4, 0, 0, y5, y8, y7]


 
2008 . Coloring, {2, 3, 4, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 2 + 56s 5 + 16s 6 - 64s 7 - 256s 8   p' = s 3 + 4s 4 - 8s 6 - 16s 7   p' = s 2 - 16s 4 - 8s 5 + 16s 6 + 64s 7

R: [7, 8, 8, 7, A, 3, B, B, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
5 vs 8 8 vs 9 8 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, -6 y5 - 3 y1 - 3 y2 - 3 y3 + 13 y4 - 3 y7 + 13 y6, 3 y5, 3 y1, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y7, 3 y6]

  p = s 4 + s 5 - s 7 - s 8   p = s 4 - s 6 - s 7 + s 9

Omega Rank for B :  cycles: {{3, 5, 7, 10, 12}}    order:   5

See Matrix
 

[y6, 0, y7, 0, y5, y3, y4, 0, 0, y1, y2, y8]


 
2009 . Coloring, {2, 3, 4, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 7, A, A, A, B, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, 2 y5, 0, 2 y4, 0, 0, -2 y5 + 11 y4 + 11 y3 - 39 y2 + 11 y1 - 2 y6, 2 y3, 2 y2, 2 y1, 2 y6, 3 y4 + 3 y3 - 11 y2 + 3 y1]

  p = - s + s 7   p' = - s + s 7

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}}    order:   5

See Matrix
 

[y1, 0, y2, 0, y5, y6, y7, 0, 0, 0, y3, y4]


 
2010 . Coloring, {2, 3, 5, 6, 7, 8, 9, 10, 11}

R: [7, 8, 8, 6, 3, 3, A, B, C, 2, 4, 5]
B: [6, 7, 7, 7, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y8, y7, y6, y5, y4, y3, y2, 0, y5, y1, y3]

  p' = s 4 - s 9   p = s 4 - s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y5, 0, 0, 0, 0, y1, y2, 0, y3, y4, y6, y7]


 
2011 . Coloring, {2, 3, 5, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 2 + 3s 3 - 6s 4 + 8s 5 - 24s 6 + 32s 7 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, C, 2, 1, 9]
B: [6, 7, 7, 7, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 8 vs 10 4 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y4 - 3 y2 + 8 y8 - 3 y7 + 5 y5, 3 y1, -3 y1 - 3 y3 + 5 y8 - 3 y6 + 8 y5, 0, 0, 3 y4, 3 y3, 3 y2, 3 y8, 3 y7, 3 y6, 3 y5]

  p' = - s 3 + s 9   p = - s 3 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {4, 7, 11}}    order:   3

See Matrix
 

[0, 0, 0, y2, -y1 + y3, y1, y4, 0, 0, y2, y3, y4]

  p = - s 2 + s 5   p' = - s 2 + s 5   p' = s 3 - s 6


M              \ ;       N

$ [ [0, 308, 332, 0, 0, 0, 60, 0, 384, 0, 120, 0] , [308, 0, 0, 0, 0, 166, 0, 286, 0, 223, 0, 221] , [332, 0, 0, 0, 0, 222, 0, 194, 0, 349, 0, 107] , [0, 0, 0, 0, 384, 60, 0, 0, 0, 332, 0, 428] , [0, 0, 0, 384, 0, 0, 378, 0, 0, 0, 442, 0] , [0, 166, 222, 60, 0, 0, 320, 0, 214, 0, 222, 0] , [60, 0, 0, 0, 378, 320, 0, 320, 0, 446, 0, 884] , [0, 286, 194, 0, 0, 0, 320, 0, 164, 0, 240, 0] , [384, 0, 0, 0, 0, 214, 0, 164, 0, 442, 0, 0] , [0, 223, 349, 332, 0, 0, 446, 0, 442, 0, 616, 0] , [120, 0, 0, 0, 442, 222, 0, 240, 0, 616, 0, 768] , [0, 221, 107, 428, 0, 0, 884, 0, 0, 0, 768, 0] ] $     $ [ [0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1] , [1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1] , [1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0] , [0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0] , [1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1] , [0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0] ] $

τ= 72 , r'= 1/2

R: [7, 8, 8, 6, 3, 3, A, B, C, 2, 1, 9]
B: [6, 7, 7, 7, A, A, B, C, B, C, 4, 5]

Ranges

Action of R on ranges, [[23], [23], [24], [25], [3], [11], [27], [7], [26], [11], [27], [7], [26], [10], [10], [6], [21], [12], [2], [12], [13], [2], [29], [8], [28], [30], [5], [9], [1], [4]]
Action of B on ranges, [[20], [20], [22], [22], [15], [24], [25], [25], [18], [24], [25], [25], [18], [24], [24], [25], [18], [29], [16], [29], [29], [16], [30], [30], [19], [30], [17], [30], [17], [14]]

Cycles:    R , {{1, 2, 7, 8, 10, 11}, {9, 12}},   B , {{5, 10, 12}, {4, 7, 11}}

β({1, 2}) = 11/344
β({1, 3}) = 83/2408
β({1, 7}) = 15/2408
β({1, 9}) = 12/301
β({1, 11}) = 15/1204
β({2, 6}) = 83/4816
β({2, 8}) = 143/4816
β({2, 10}) = 223/9632
β({2, 12}) = 221/9632
β({3, 6}) = 111/4816
β({3, 8}) = 97/4816
β({3, 10}) = 349/9632
β({3, 12}) = 107/9632
β({4, 5}) = 12/301
β({4, 6}) = 15/2408
β({4, 10}) = 83/2408
β({4, 12}) = 107/2408
β({5, 7}) = 27/688
β({5, 11}) = 221/4816
β({6, 7}) = 10/301
β({6, 9}) = 107/4816
β({6, 11}) = 111/4816
β({7, 8}) = 10/301
β({7, 10}) = 223/4816
β({7, 12}) = 221/2408
β({8, 9}) = 41/2408
β({8, 11}) = 15/602
β({9, 10}) = 221/4816
β({10, 11}) = 11/172
β({11, 12}) = 24/301

Partitions
α([{2, 3, 4, 7, 9, 11}, {1, 5, 6, 8, 10, 12}]) = 1/1

b1 = {2, 3, 4, 7, 9, 11} ` , ` b2 = {1, 5, 6, 8, 10, 12}

Action of R and B on the blocks of the partitions: = [2, 1] [1, 2]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {2, 3, 5, 6, 7, 8, 9, 10, 12}
Rank2
R,B [7, 8, 8, 6, 3, 3, A, B, C, 2, 1, 9], [6, 7, 7, 7, A, A, B, C, B, C, 4, 5]
π2 [308, 332, 0, 0, 0, 60, 0, 384, 0, 120, 0, 0, 0, 0, 166, 0, 286, 0, 223, 0, 221, 0, 0, 222, 0, 194, 0, 349, 0, 107, 384, 60, 0, 0, 0, 332, 0, 428, 0, 378, 0, 0, 0, 442, 0, 320, 0, 214, 0, 222, 0, 320, 0, 446, 0, 884, 164, 0, 240, 0, 442, 0, 0, 616, 0, 768]
u2 [1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1] (dim 1)
wpp [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]

 

 
2012 . Coloring, {2, 3, 5, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 2 - 3s 3 - 6s 4 - 8s 5 - 24s 6 - 32s 7 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, C, C, 4, 9]
B: [6, 7, 7, 7, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 10 9 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 8, 11}}    order:   10

See Matrix
 

[0, 0, -y1 - y2 + 2 y4 + 2 y5 - y3 - y6, y1, 0, y2, y4 + y5 - y7, y3, y4, y5, y6, y7]

  p' = - s 3 + s 8   p = - s 3 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1 + y2 + y3 - y4 - y5 + y6 - y7, y1, 0, 0, y2, y3, y4, 0, 0, y5, y6, y7]

  p = - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


M              \ ;       N

$ [ [0, 956, 0, 0, 1260, 1164, 0, 0, 0, 0, 2328, 0] , [956, 0, 0, 0, 0, 0, 2328, 0, 0, 1539, 0, 885] , [0, 0, 0, 0, 0, 0, 1212, 0, 1284, 1305, 0, 1907] , [0, 0, 0, 0, 0, 0, 1164, 0, 1260, 1812, 0, 1472] , [1260, 0, 0, 0, 0, 0, 2132, 0, 0, 1284, 0, 1032] , [1164, 0, 0, 0, 0, 0, 478, 0, 736, 1794, 0, 1536] , [0, 2328, 1212, 1164, 2132, 478, 0, 478, 0, 0, 3624, 0] , [0, 0, 0, 0, 0, 0, 478, 0, 1396, 1770, 0, 2064] , [0, 0, 1284, 1260, 0, 736, 0, 1396, 0, 0, 1032, 0] , [0, 1539, 1305, 1812, 1284, 1794, 0, 1770, 0, 0, 1912, 0] , [2328, 0, 0, 0, 0, 0, 3624, 0, 1032, 1912, 0, 2520] , [0, 885, 1907, 1472, 1032, 1536, 0, 2064, 0, 0, 2520, 0] ] $     $ [ [0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0] , [0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0] , [1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1] , [0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0] ] $

τ= 74 , r'= 1/2

R: [7, 8, 8, 6, 3, 3, A, B, C, C, 4, 9]
B: [6, 7, 7, 7, A, A, B, C, B, 2, 1, 5]

Ranges

Action of R on ranges, [[23], [8], [8], [12], [26], [27], [25], [26], [27], [27], [25], [21], [22], [22], [20], [10], [11], [9], [10], [11], [11], [9], [29], [14], [30], [30], [28], [15], [15], [13]]
Action of B on ranges, [[19], [21], [21], [3], [24], [5], [16], [24], [24], [5], [16], [24], [24], [5], [16], [29], [6], [17], [29], [29], [6], [17], [30], [4], [30], [7], [18], [4], [1], [2]]

Cycles:    R , {{9, 12}, {3, 4, 6, 8, 11}},   B , {{1, 2, 6, 7, 10, 11}}

β({1, 2}) = 239/11416
β({1, 5}) = 315/11416
β({1, 6}) = 291/11416
β({1, 11}) = 291/5708
β({2, 7}) = 291/5708
β({2, 10}) = 1539/45664
β({2, 12}) = 885/45664
β({3, 7}) = 303/11416
β({3, 9}) = 321/11416
β({3, 10}) = 1305/45664
β({3, 12}) = 1907/45664
β({4, 7}) = 291/11416
β({4, 9}) = 315/11416
β({4, 10}) = 453/11416
β({4, 12}) = 46/1427
β({5, 7}) = 533/11416
β({5, 10}) = 321/11416
β({5, 12}) = 129/5708
β({6, 7}) = 239/22832
β({6, 9}) = 23/1427
β({6, 10}) = 897/22832
β({6, 12}) = 48/1427
β({7, 8}) = 239/22832
β({7, 11}) = 453/5708
β({8, 9}) = 349/11416
β({8, 10}) = 885/22832
β({8, 12}) = 129/2854
β({9, 11}) = 129/5708
β({10, 11}) = 239/5708
β({11, 12}) = 315/5708

Partitions
α([{1, 7, 9, 10, 12}, {2, 3, 4, 5, 6, 8, 11}]) = 1/1

b1 = {1, 7, 9, 10, 12} ` , ` b2 = {2, 3, 4, 5, 6, 8, 11}

Action of R and B on the blocks of the partitions: = [1, 2] [2, 1]
with invariant measure [1, 1]

N by blocks, check: true . ` See partition graph.

` ` See level-2 partition graph.

`

Right Group
Coloring {2, 3, 5, 6, 7, 8, 9, 11, 12}
Rank2
R,B [7, 8, 8, 6, 3, 3, A, B, C, C, 4, 9], [6, 7, 7, 7, A, A, B, C, B, 2, 1, 5]
π2 [956, 0, 0, 1260, 1164, 0, 0, 0, 0, 2328, 0, 0, 0, 0, 0, 2328, 0, 0, 1539, 0, 885, 0, 0, 0, 1212, 0, 1284, 1305, 0, 1907, 0, 0, 1164, 0, 1260, 1812, 0, 1472, 0, 2132, 0, 0, 1284, 0, 1032, 478, 0, 736, 1794, 0, 1536, 478, 0, 0, 3624, 0, 1396, 1770, 0, 2064, 0, 1032, 0, 1912, 0, 2520]
u2 [1, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1] (dim 1)
wpp [5, 7, 7, 7, 7, 7, 5, 7, 5, 5, 7, 5]

 

 
2013 . Coloring, {2, 3, 5, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, 3, A, B, B, 2, 4, 9]
B: [6, 7, 7, 7, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}}    order:   5

See Matrix
 

[0, y1, y2, y3, 0, y4, y5, y6, 2 y5, y7, y8, 0]

  p = - s 4 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y7, 0, 0, 0, y6, y5, y4, 0, 0, y3, y2, y1]


 
2014 . Coloring, {2, 3, 5, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 8, 8, 6, 3, 3, A, C, C, 2, 4, 9]
B: [6, 7, 7, 7, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 8 vs 9 8 vs 9 6 vs 9 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y2, y1, 8 y2 - 4 y1 - 6 y3, 0, y3, 4 y2 - 2 y1 - 3 y3, y4, y5, 6 y2 - 3 y1 - 4 y3, 0, y6]

  p = s 5 - s 9   p' = - s 5 + s 7   p' = s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y7, y5, y6, 0, 0, y2, y3, y4]


 
2015 . Coloring, {2, 3, 5, 6, 8, 9, 10, 11, 12}

R: [7, 8, 8, 6, 3, 3, B, B, C, 2, 4, 9]
B: [6, 7, 7, 7, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 6 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, 6 y6, -3 y1 - 3 y7 - 9 y6 - 3 y4 + 13 y5 - 3 y3 + 13 y2, 3 y1, 0, 3 y7, 3 y6, 3 y4, 3 y5, 0, 3 y3, 3 y2]

  p = - s 2 - s 3 + s 7 + s 8   p = s 2 - s 4 - s 7 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y3, y2, 3 y4, 0, 0, y5, y4, y6]

  p = - s 4 + s 7


 
2016 . Coloring, {2, 3, 5, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 2s 2 - 7s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 8, 6, 3, A, A, B, C, 2, 4, 9]
B: [6, 7, 7, 7, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 10 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 6, 8, 10, 11}}    order:   6

See Matrix
 

[0, 3 y2, 3 y1, 3 y3, 0, -3 y2 - 6 y1 + 5 y5 - 3 y7 + 8 y6, 3 y1, -3 y3 + 8 y5 - 3 y4 + 5 y6, 3 y5, 3 y4, 3 y7, 3 y6]

  p = - s 2 + s 8   p' = - s 2 + s 8   p' = - s 3 + s 9

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[-3 y1 + 5 y2 - 3 y4 - 3 y3 + 5 y5 - 3 y7 + 5 y6, 0, 3 y1, 0, 3 y2, 3 y4, 3 y3, 0, 0, 3 y5, 3 y7, 3 y6]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
2017 . Coloring, {2, 3, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 6s 2 + 5s 3 - 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 6, A, 3, A, B, C, 2, 4, 9]
B: [6, 7, 7, 7, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 8

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y8 - 3 y4 + 13 y5 - 3 y6 - 3 y9 + 13 y7, 3 y1, 3 y2, 0, 3 y3, 3 y8, 3 y4, 3 y5, 3 y6, 3 y9, 3 y7]

  p = - s 4 - s 5 + s 9 + s 10

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1 - y4 + y2 - y3 - y5 + y6 + y7, 0, y1, 0, y4, y2, y3, 0, 0, y5, y6, y7]

  p = - s + s 2 - s 3 + s 4 - s 5 + s 6 - s 7 + s 8


 
2018 . Coloring, {2, 4, 5, 6, 7, 8, 9, 10, 11}

R: [7, 8, 7, 7, 3, 3, A, B, C, 2, 4, 5]
B: [6, 7, 8, 6, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1 + y2 - y3 - y4 + y5 + y6 - y7 + y8, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

  p = - s 4 + s 5 - s 6 + s 7 - s 8 + s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y4, 0, 0, 0, 0, y3, y2, y2, -y4 + y3 - 2 y2 - y1 + y6 + y5, y1, y6, y5]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7   p = - s 2 + s 8


 
2019 . Coloring, {2, 4, 5, 6, 7, 8, 9, 10, 12}

R: [7, 8, 7, 7, 3, 3, A, B, C, 2, 1, 9]
B: [6, 7, 8, 6, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y4 - 3 y3 + 8 y1 - 3 y2 + 5 y7, -3 y5 + 5 y1 - 3 y6 + 8 y7, 3 y4, 0, 0, 0, 3 y5, 3 y3, 3 y1, 3 y2, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y2, y1, y7, y6, y6, 0, y5, y4, y3]

  p = - s 5 + s 8


 
2020 . Coloring, {2, 4, 5, 6, 7, 8, 9, 11, 12}

R: [7, 8, 7, 7, 3, 3, A, B, C, C, 4, 9]
B: [6, 7, 8, 6, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, 0, 2 y3, y1, 0, 0, y2, y3, y4, y7, y5, y6]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y4, y5, 0, 0, y3, y2, y1, y9, 0, y7, y8, y6]


 
2021 . Coloring, {2, 4, 5, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 + 32s 8

R: [7, 8, 7, 7, 3, 3, A, B, B, 2, 4, 9]
B: [6, 7, 8, 6, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y1, y5, y2, 0, 0, y7, y6, y5, y4, y3, 0]

  p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, y4, 0, y5, y6, y7]

  p = - s 5 + s 8


 
2022 . Coloring, {2, 4, 5, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, 3, A, C, C, 2, 4, 9]
B: [6, 7, 8, 6, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, y1, y2, y2, 0, 0, y3, y4, y5, y6, 0, y7]

  p = - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y5, 0, 0, 0, y1, y2, y3, y3, 0, y4, y6, y7]

  p = - s 5 + s 8


 
2023 . Coloring, {2, 4, 5, 6, 8, 9, 10, 11, 12}

R: [7, 8, 7, 7, 3, 3, B, B, C, 2, 4, 9]
B: [6, 7, 8, 6, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1, 3 y1, -6 y1 - 3 y2 - 3 y4 + 13 y3 - 3 y5 + 13 y6, 0, 0, 3 y2, 3 y4, 3 y3, 0, 3 y5, 3 y6]

  p = s 3 + s 4 - s 6 - s 7   p = - s 3 + s 5 + s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y6, y6, 0, y5, y6, y4]

  p' = s 4 - s 7   p = s 4 - s 7


 
2024 . Coloring, {2, 4, 5, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 5 + 8s 6 - 32s 8

R: [7, 8, 7, 7, 3, A, A, B, C, 2, 4, 9]
B: [6, 7, 8, 6, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y1, 3 y2, -3 y2 - 3 y3 + 8 y4 - 3 y5 + 5 y7, 0, 0, -3 y1 + 5 y4 - 3 y6 + 8 y7, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p' = - s 2 + s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   9

See Matrix
 

[y8, 0, y9, 0, y1, y2, y3, y4, 0, y5, y6, y7]


 
2025 . Coloring, {2, 4, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 6 - 32s 8

R: [7, 8, 7, 7, A, 3, A, B, C, 2, 4, 9]
B: [6, 7, 8, 6, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 7 vs 9 9 vs 9

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y2 + 5 y3 - 3 y5 + 8 y7, -3 y1 - 3 y6 + 8 y3 - 3 y4 + 5 y7, 3 y1, 0, 0, 3 y2, 3 y6, 3 y3, 3 y4, 3 y5, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{3, 5, 8, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y5, y6, y3, y4, 0, y7, y8, y9]


 
2026 . Coloring, {2, 5, 6, 7, 8, 9, 10, 11, 12}

R: [7, 8, 7, 6, 3, 3, A, B, C, 2, 4, 9]
B: [6, 7, 8, 7, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 10 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 3, 4, 6, 7, 8, 10, 11}}    order:   8

See Matrix
 

[0, 3 y2, 3 y3, -3 y3 - 3 y4 + 8 y7 - 3 y8 + 5 y6, 0, -3 y2 - 3 y1 + 5 y7 - 3 y5 + 8 y6, 3 y1, 3 y4, 3 y7, 3 y8, 3 y5, 3 y6]

  p' = - s + s 9   p = - s + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y5, 0, 0, 0, y6, y4, 2 y1, y1, 0, y2, y3, y7]

  p = - s 5 + s 8


 
2027 . Coloring, {3, 4, 5, 6, 7, 8, 9, 10, 11}

R: [7, 7, 8, 7, 3, 3, A, B, C, 2, 4, 5]
B: [6, 8, 7, 6, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 9 6 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   9

See Matrix
 

[0, y1, y2, y3, y4, 0, y5, y6, 0, y7, y8, y9]

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y6 - 2 y5 - y4 - y3 + y2 + y1, 0, 0, 0, 0, y6, y5, y5, y4, y3, y2, y1]

  p = - s 2 + s 8   p = s 2 - s 3 + s 4 - s 5 + s 6 - s 7


 
2028 . Coloring, {3, 4, 5, 6, 7, 8, 9, 10, 12}

R: [7, 7, 8, 7, 3, 3, A, B, C, 2, 1, 9]
B: [6, 8, 7, 6, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[3 y2, 3 y1, -3 y2 - 3 y1 - 3 y8 - 3 y7 + 13 y6 - 3 y5 - 3 y4 + 13 y3, 0, 0, 0, 3 y8, 3 y7, 3 y6, 3 y5, 3 y4, 3 y3]

  p = s 5 + s 6 - s 8 - s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y5, y6, y3, y3, 0, y1, y2, y7]

  p = s 5 - s 8


 
2029 . Coloring, {3, 4, 5, 6, 7, 8, 9, 11, 12}

R: [7, 7, 8, 7, 3, 3, A, B, C, C, 4, 9]
B: [6, 8, 7, 6, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 8 9 vs 9

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y5, y4, 0, 0, y3, y1, y2, y8, y6, y7]

Omega Rank for B :  cycles: {{2, 5, 8, 10, 12}}    order:   5

See Matrix
 

[y2, y1, 0, 0, y7, y5, y6, y4, 0, y3, y9, y8]


 
2030 . Coloring, {3, 4, 5, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 6 - 32s 8

R: [7, 7, 8, 7, 3, 3, A, B, B, 2, 4, 9]
B: [6, 8, 7, 6, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}}    order:   6

See Matrix
 

[0, y3, y4, y2, 0, 0, y1, y5, y4, y7, y6, 0]

  p = s 5 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y6, y6, 0, y7, y4, y5]

  p = - s 5 + s 8


 
2031 . Coloring, {3, 4, 5, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 3 + 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, 3, A, C, C, 2, 4, 9]
B: [6, 8, 7, 6, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 7 vs 8

Omega Rank for R :  cycles: {{2, 7, 10}, {9, 12}}    order:   6

See Matrix
 

[0, 2 y1, 7 y1 + 7 y6 - 9 y5 - 9 y4 + 7 y3 - 9 y2, 7 y1 + 7 y6 - 9 y5 - 9 y4 + 7 y3 - 9 y2, 0, 0, 2 y6, 2 y5, 2 y4, 2 y3, 0, 2 y2]

  p = - s 3 - s 4 + s 6 + s 7   p' = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y3, y2, y7, y7, 0, y5, y6, y4]

  p = - s 5 + s 8


 
2032 . Coloring, {3, 4, 5, 6, 8, 9, 10, 11, 12}

R: [7, 7, 8, 7, 3, 3, B, B, C, 2, 4, 9]
B: [6, 8, 7, 6, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 8 6 vs 8

Omega Rank for R :  cycles: {{4, 7, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y1, 3 y1, 3 y6, 0, 0, -6 y1 - 3 y6 - 3 y3 + 13 y5 - 3 y4 + 13 y2, 3 y3, 3 y5, 0, 3 y4, 3 y2]

  p = - s 3 - s 4 + s 6 + s 7   p = s 3 - s 5 - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, y1, y5, y6, y6, 0, y4, y6, y3]

  p = - s 4 + s 7   p' = - s 4 + s 7


 
2033 . Coloring, {3, 4, 5, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 + 32s 8

R: [7, 7, 8, 7, 3, A, A, B, C, 2, 4, 9]
B: [6, 8, 7, 6, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 8 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y2 - 3 y1 - 3 y3 - 3 y4 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y2, 3 y1, 0, 0, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}, {1, 3, 6, 7, 11}}   

See Matrix
 

[7 y1, 0, -7 y1 + 9 y5 - 7 y4 - 7 y3 + 9 y2 + 9 y8 - 7 y7 + 9 y6, 0, 7 y5, 7 y4, 7 y3, 7 y2, 0, 7 y8, 7 y7, 7 y6]

  p = s 2 + s 3 + s 4 - s 7 - s 8 - s 9


 
2034 . Coloring, {3, 4, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 - 8s 5 - 8s 6 + 32s 8

R: [7, 7, 8, 7, A, 3, A, B, C, 2, 4, 9]
B: [6, 8, 7, 6, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 10 vs 10 10 vs 10 8 vs 9 9 vs 9

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y4 - 3 y2 - 3 y3 + 13 y5 - 3 y6 - 3 y7 + 13 y8, 3 y1, 3 y4, 0, 0, 3 y2, 3 y3, 3 y5, 3 y6, 3 y7, 3 y8]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y4, 0, y2, y3, y6, y5, 0, y7, y9, y8]


 
2035 . Coloring, {3, 5, 6, 7, 8, 9, 10, 11, 12}

R: [7, 7, 8, 6, 3, 3, A, B, C, 2, 4, 9]
B: [6, 8, 7, 7, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 6 vs 10 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {3, 4, 6, 8, 11}, {2, 7, 10}}   

See Matrix
 

[0, 2 y4, 7 y4 - 2 y1 - 2 y2 - 2 y3 - 2 y5, 2 y1, 0, 2 y2, 2 y4, 2 y3, 3 y4 - 2 y6, 2 y4, 2 y5, 2 y6]

  p = - s - s 2 + s 6 + s 7   p = s - s 3 - s 6 + s 8   p = - s - s 4 + s 6 + s 9   p = s - s 5 - s 6 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y4, 0, 0, 0, y3, y2, 2 y1, y1, 0, y6, y5, y7]

  p = s 5 - s 8


 
2036 . Coloring, {4, 5, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 6 - 32s 8

R: [7, 7, 7, 7, 3, 3, A, B, C, 2, 4, 9]
B: [6, 8, 8, 6, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y4 + 13 y3 - 3 y2 - 9 y5 + 13 y6, 6 y5, 3 y1, 0, 0, 3 y4, 0, 3 y3, 3 y2, 3 y5, 3 y6]

  p = - s 3 - s 4 + s 6 + s 7   p = - s 3 + s 5 + s 6 - s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[2 y1, 0, 0, 0, 2 y2, 2 y3, 0, 2 y4, 0, 2 y5, 3 y4, 2 y6]

  p = s 4 - s 7


 
2037 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Ωp(Δ)=0:     p = s 3 - 2s 4 - 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, 2, 4, 5]
B: [6, 7, 7, 6, A, A, B, C, B, C, 1, 9]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 8 vs 9 6 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -y1 + y2 + y3 - y4 + y5 + y6 - y7 - y8, y1, y2, y3, 0, y4, y5, 0, y6, y7, y8]

  p = s 4 - s 5 + s 6 - s 7 + s 8 - s 9

Omega Rank for B :  cycles: {{1, 6, 9, 10, 11, 12}}    order:   6

See Matrix
 

[y1 - y2 - y3 - y4 + y5 + y6, 0, 0, 0, 0, y1, y2, 0, y3, y4, y5, y6]

  p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7


 
2038 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 10, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, 2, 1, 9]
B: [6, 7, 7, 6, A, A, B, C, B, C, 4, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 9 7 vs 7

Omega Rank for R :  cycles: {{1, 2, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[-3 y3 + 5 y4 - 3 y5 + 8 y7, -3 y1 - 3 y2 + 8 y4 - 3 y6 + 5 y7, 3 y1, 0, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[0, 0, 0, y4, y1, y2, y3, 0, 0, y7, y5, y6]


 
2039 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 11, 12}

Ωp(Δ)=0:     p = s 3 + 2s 4 + 8s 5 + 8s 6 - 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, C, 4, 9]
B: [6, 7, 7, 6, A, A, B, C, B, 2, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 8 8 vs 8

Omega Rank for R :  cycles: {{9, 12}}    order:   8

See Matrix
 

[0, 0, y1, y1 + y5 - y2 - y3 - y4 + y7 + y6, 0, 0, y5, y2, y3, y4, y7, y6]

  p = - s 7 + s 8

Omega Rank for B :  cycles: {{1, 2, 6, 7, 10, 11}}    order:   6

See Matrix
 

[y1, y2, 0, 0, y3, y4, y5, 0, 0, y6, y7, y8]


 
2040 . Coloring, {2, 3, 4, 5, 6, 7, 8, 10, 11, 12}

Ωp(Δ)=0:     p = 3s 2 + 10s 3 + 40s 5 + 32s 6 + 32s 7 + 128s 8   p = 9s 2 + 20s 4 + 40s 5 + 16s 6 + 96s 7 + 64s 8

R: [7, 8, 8, 7, 3, 3, A, B, B, 2, 4, 9]
B: [6, 7, 7, 6, A, A, B, C, C, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, y3, y4, y3 + y1 - y2 - y5 + y6, 0, 0, y1, y2, y4, y5, y6, 0]

  p = - s 2 + s 8   p = - s 2 + s 3 - s 4 + s 5 - s 6 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y2, 0, 0, 0, y1, y5, y3, 0, 0, y4, y6, y7]


 
2041 . Coloring, {2, 3, 4, 5, 6, 7, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 - 8s 4 + 8s 5 - 32s 7   p = s 2 - 12s 4 - 8s 5 + 16s 6 - 32s 7 - 64s 8

R: [7, 8, 8, 7, 3, 3, A, C, C, 2, 4, 9]
B: [6, 7, 7, 6, A, A, B, B, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 8 vs 8 8 vs 8 6 vs 8 7 vs 7

Omega Rank for R :  cycles: {{9, 12}}    order:   6

See Matrix
 

[0, -y2 + y3 + y6 + y4 - y5, y1, y1, 0, 0, y2, y3, y6, y4, 0, y5]

  p = - s 6 + s 7   p = - s 6 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, y4, 0, 0, y5, y6, y7]


 
2042 . Coloring, {2, 3, 4, 5, 6, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 8s 6 - 16s 7   p' = s 3 - 8s 6 - 16s 7

R: [7, 8, 8, 7, 3, 3, B, B, C, 2, 4, 9]
B: [6, 7, 7, 6, A, A, A, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 6 vs 8 6 vs 7

Omega Rank for R :  cycles: {{9, 12}, {4, 7, 11}}    order:   6

See Matrix
 

[0, 3 y1, 3 y1, -6 y1 - 3 y2 - 3 y3 + 13 y4 - 3 y5 + 13 y6, 0, 0, 3 y2, 3 y3, 3 y4, 0, 3 y5, 3 y6]

  p = - s 3 + s 5 + s 6 - s 8   p' = - s 3 - s 4 + s 6 + s 7

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y3, 2 y5, 0, 0, y4, y5, y6]

  p = - s 4 + s 7


 
2043 . Coloring, {2, 3, 4, 5, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p' = s 2 - 2s 3 + 8s 4 - 8s 5 + 32s 7   p = s 2 + 4s 4 + 8s 5 - 16s 6 + 32s 7 + 64s 8

R: [7, 8, 8, 7, 3, A, A, B, C, 2, 4, 9]
B: [6, 7, 7, 6, A, 3, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y7, 3 y6, 3 y5, 0, 0, 3 y4, 3 y3, 3 y2, -3 y5 - 3 y3 + 8 y2 + 5 y1, -3 y7 - 3 y6 - 3 y4 + 5 y2 + 8 y1, 3 y1]

  p' = s 2 - s 8   p = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 6, 7, 11}, {5, 10, 12}}   

See Matrix
 

[-3 y3 + 5 y1 - 3 y2 - 3 y4 + 5 y6 - 3 y7 + 5 y5, 0, 3 y3, 0, 3 y1, 3 y2, 3 y4, 0, 0, 3 y6, 3 y7, 3 y5]

  p = - s - s 2 - s 3 + s 6 + s 7 + s 8


 
2044 . Coloring, {2, 3, 4, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = - 9s 2 + 4s 4 - 24s 5 - 16s 6 + 96s 7 + 64s 8   p = - 3s 2 + 2s 3 - 8s 5 + 32s 7

R: [7, 8, 8, 7, A, 3, A, B, C, 2, 4, 9]
B: [6, 7, 7, 6, 3, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
6 vs 8 9 vs 9 9 vs 9 7 vs 9 8 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, 3 y1, -3 y1 - 3 y2 + 5 y4 - 3 y6 + 8 y7, -3 y3 + 8 y4 - 3 y5 + 5 y7, 0, 0, 3 y2, 3 y3, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{1, 3, 5, 6, 7, 10, 11, 12}}    order:   8

See Matrix
 

[y1, 0, y2, 0, y7, y3, y4, 0, 0, y6, y8, y5]


 
2045 . Coloring, {2, 3, 5, 6, 7, 8, 9, 10, 11, 12}

R: [7, 8, 8, 6, 3, 3, A, B, C, 2, 4, 9]
B: [6, 7, 7, 7, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 9 vs 10 7 vs 7

Omega Rank for R :  cycles: {{3, 4, 6, 8, 11}, {9, 12}}    order:   10

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y6 - 3 y7 + 13 y8 - 3 y4 - 3 y9 + 13 y5, 3 y1, 3 y2, 0, 3 y3, 3 y6, 3 y7, 3 y8, 3 y4, 3 y9, 3 y5]

  p = - s 4 - s 5 + s 9 + s 10

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y6, y5, y4, 0, 0, y2, y3, y7]


 
2046 . Coloring, {2, 4, 5, 6, 7, 8, 9, 10, 11, 12}

R: [7, 8, 7, 7, 3, 3, A, B, C, 2, 4, 9]
B: [6, 7, 8, 6, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 7 vs 9 7 vs 8

Omega Rank for R :  cycles: {{2, 4, 7, 8, 10, 11}, {9, 12}}    order:   6

See Matrix
 

[0, -3 y2 + 5 y7 - 3 y4 + 8 y5, -3 y1 - 3 y3 + 8 y7 - 3 y6 + 5 y5, 3 y1, 0, 0, 3 y2, 3 y3, 3 y7, 3 y6, 3 y4, 3 y5]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y2, y7, y6, y6, 0, y5, y4, y3]

  p = - s 5 + s 8


 
2047 . Coloring, {3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

R: [7, 7, 8, 7, 3, 3, A, B, C, 2, 4, 9]
B: [6, 8, 7, 6, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
8 vs 8 10 vs 10 10 vs 10 8 vs 9 7 vs 8

Omega Rank for R :  cycles: {{9, 12}, {2, 7, 10}}    order:   6

See Matrix
 

[0, -3 y1 - 3 y2 - 3 y3 - 3 y8 + 13 y6 - 3 y7 - 3 y4 + 13 y5, 3 y1, 3 y2, 0, 0, 3 y3, 3 y8, 3 y6, 3 y7, 3 y4, 3 y5]

  p = - s 5 - s 6 + s 8 + s 9

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y1, 0, 0, 0, y7, y6, y5, y5, 0, y4, y3, y2]

  p = - s 5 + s 8


 
2048 . Coloring, {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Ωp(Δ)=0:     p = s 3 - 2s 4 + 8s 5 - 8s 6 + 32s 8

R: [7, 8, 8, 7, 3, 3, A, B, C, 2, 4, 9]
B: [6, 7, 7, 6, A, A, B, C, B, C, 1, 5]

` See graph

` ` See pair graph

`
Δ-RankA+(1/2)Δ A-(1/2)ΔRB
7 vs 8 9 vs 9 9 vs 9 7 vs 9 7 vs 7

Omega Rank for R :  cycles: {{9, 12}, {2, 4, 7, 8, 10, 11}}    order:   6

See Matrix
 

[0, -3 y2 - 3 y1 + 8 y4 - 3 y6 + 5 y7, 3 y2, 3 y3, 0, 0, 3 y1, -3 y3 + 5 y4 - 3 y5 + 8 y7, 3 y4, 3 y5, 3 y6, 3 y7]

  p = - s 2 + s 8   p' = - s 2 + s 8

Omega Rank for B :  cycles: {{5, 10, 12}}    order:   6

See Matrix
 

[y4, 0, 0, 0, y3, y2, y1, 0, 0, y7, y6, y5]


 
SUMMARY
Graph Type
CC
ν(A)
 4
ν(Δ)
 4
π
 [1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2]
Dbly Stoch
false

 
SANDWICH
Total 0
No .ColoringRank

 
RT GROUPS
Total 12
No .ColoringRankSolv
1 {2, 3, 5, 6, 7, 8, 9, 11, 12} 2 Not Solvable
2 {2, 3, 5, 6, 7, 8, 9, 10, 12} 2 Not Solvable
3 {2, 3, 4, 5, 6, 7, 8, 11} 8 Not Solvable
4 {2, 3, 6, 7, 8, 10} 2 Not Solvable
5 {2, 3, 6, 7, 8, 11} 2 Not Solvable
6 {5, 9, 12} 2 Not Solvable
7 {2, 3, 4, 9, 10} 8 Not Solvable
8 {10, 11} 2 Not Solvable
9 {2, 3, 4, 9, 10, 11} 8 Not Solvable
10 {5, 9, 10, 11, 12} 2 Not Solvable
11 {} 2 Not Solvable
12 {2, 3, 4, 5, 6, 7, 8} 8 Not Solvable

 

Δ-RANK'DSC'D !RK'D τ-RANK'DR/B RANK'DNOT SYNC'D Total Runs2n-1
998 60 1935 , 1927 433 , 585 12 2048 2048