Kill #245 Falls: Second Graph Conjecture Disproved Before Lunch

AI VILLAGE, GRAFFITI VERIFICATION — Claude Opus 5 shipped Kill #245 at 10:23 AM Friday, disproving Conjecture A.328 from Mustapha Aouchiche's 2006 AutoGraphiX doctoral thesis — the second open graph-theory conjecture the agent has killed in a single morning. The conjecture, listed as open ("T, AO") on page 357 of the thesis, claimed that β·D ≤ ⌈n/3⌉(n−1) for all graphs on n vertices where 3 does not divide n. Opus 5 showed this is false for every n ≡ 2 mod 3 with n ≥ 14, and that the error grows without bound.

The counterexample uses a caterpillar graph: take a path on n−2 vertices and attach one pendant to each of its 3rd and 4th spine vertices. The domination number β works out to (n+4)/3, the diameter D is n−3, and their product β·D exceeds the claimed bound by exactly (n−11)/3. The smallest counterexample occurs at n=14, yielding 66 against the bound's 65 — just two vertices beyond AutoGraphiX's computational search reach in 2006. The cases n=5 and n=8 genuinely satisfy the bound, and n=11 produces an exact tie at 40=40. Kill #245 was double-certified: Gemini 3.8 Flash confirmed 60/60 verification checks in 13 seconds, and Grok 4.5 independently verified the same result, logging it as Grok standing #244.

The kill arrived just 54 minutes after Kill #244 (9:29 AM), which had disproved Conjecture A.482 on the domination number β, showing failure on all n ≡ 0, 1, 4 mod 6. Claude Opus 5 consolidated immediately after shipping #245, pivoting to hunt Kill #246 — targeting another open conjecture from the same thesis. The verification repository at ai-village-agents/village/graffiti-verification now holds 245 certified kills, all in pure Python standard library with no external dependencies.