The two coloring of the edges of K57 which established the lower bound is saved here as a 0-1 adjacency matrix .
The construction requires three steps: first a set of five 19 by 19 circulant matrices is defined. Next, these five circulants are used in a 3 by 3 array to determine a coloring which is not good. Finally, 141 edges are recolored, giving the desired coloring which established the bound.
The five circulants are:
A = 19(2,3,6,13,16,17)
B = 19(3,4,6,9,10,13,15,16)
C = 19(2,4,6,7,8,10,11,12,13,15,17)
D = 19(6,7,11,13,15,16,17)
E = 19(3,4,10,12,13,14,16,18)
These five 19 by 19 blocks are used as follows to form a 57 by 57 symmetric adjacency matrix :
A C D
C A E
D E B
The 141 edges whose colors need to be changed are listed below:
0-16 7-47 12-52 18-48 29-43 45-50
0-26 8-10 13-16 18-54 29-50
0-33 8-35 13-20 19-22 30-44
0-40 8-42 13-27 19-46 31-39
1-27 8-51 13-53 20-38 32-40
1-34 8-54 14-48 20-47 33-41
1-41 9-20 15-17 20-51 33-45
1-54 9-27 15-23 21-23 33-51
2-20 10-21 15-39 21-39 34-42
2-32 10-24 15-42 21-45 34-46
2-35 10-27 15-49 21-52 34-52
2-55 10-28 15-52 22-49 34-55
3-21 10-40 16-27 23-44 35-40
3-39 10-43 16-34 23-47 35-47
3-56 10-46 16-56 23-56 35-49
4-22 10-50 17-24 24-38 35-53
4-38 10-53 17-28 25-28 36-50
4-40 10-56 17-31 25-52 37-45
5-19 11-22 17-34 26-53 37-51
5-31 11-25 17-35 27-29 38-52
5-41 11-29 17-38 27-39 39-44
5-45 11-37 17-41 27-45 40-44
5-54 11-41 17-44 27-48 40-48
6-20 11-47 17-47 27-51 40-54
6-32 11-51 17-53 27-54 41-45
6-46 12-19 18-25 28-31 41-55
7-21 12-26 18-32 28-55 42-53
7-33 12-48 18-39 29-40 44-52
Geoff Exoo