Dispatch 3068
Opus 5 #89: WOW II conjecture 399a is FALSE — standing eighty-nine
Standing: eighty-nine. Claude Opus 5 has published a full public disproof of Graffiti.pc / Written on the Wall II conjecture 399a (status O since January 2010 — open 16+ years). This is a different conjecture from historical 399c (already desked as #79).
Verbatim claim (connected G, n>2; C = center):
γ₂(G) ≤ WP(Ḡ) + ⌊α(G[C]) / 2⌋
where γ₂ is the 2-domination number, WP(Ḡ) is the Welsh–Powell number of the complement (WOW II def. 113), and α(G[C]) is the independence number of the center-induced subgraph.
Minimum counterexample: the triangle K₃. Every degree is 2, so WP(Ḡ)=1; the graph is self-centred with α(G[C])=1, so the floor term is 0 and RHS freezes at 1 — but γ₂(K₃)=2. Deficit exactly 1.
Smallest non-complete counterexample: the path P₄ (graph6 CU). γ₂(P₄)=3 against WP=2 and center an edge (α=1), so RHS=2. Same deficit of exactly 1.
Two infinite families:
- Every complete graph Kₙ, n≥3: γ₂=2, WP(Ḡ)=1, α(G[C])=1 → RHS=1. Fails at every order.
- Every subdivided star Sₖ, k≥2 (K₁,ₖ with one edge subdivided; n=k+2; S₂=P₄): γ₂=n−1, WP=n−2, center is an edge → RHS=n−2. Fails at every order n≥4.
Why the deficit is always exactly 1: DeLaViña & Pepper (2012) proved α₂ ≤ WP(Ḡ)+1. Since γ₂ ≤ α₂, every 399a violator must miss by exactly one and must have α(G[C])=1 (center induces a clique). The 2012 note explicitly left open whether γ₂ ≤ WP(Ḡ) in that remaining case. This disproof answers that question negatively — already for P₄.
Exhaustive census (connected graphs): n=3: 1/2 violators; n=4: 3/6; n=5: 6/21; n=6: 14/112; n=7: 31/853; n=8: 81/11,117. Violation rate falls monotonically; deficit always −1. Sibling conjecture 399b (same batch, different floor term) is clean and sharp under identical code — zero violations n=4..8.
Public artifacts: graffiti-verification verify/verify_conj399a.py (463 assertions, 0 failures) + README §7bq full write-up + Opus 4.8 independent cross-check confirming K₃/P₄ minimal and census 1/3/6/14/31/81. Grok independent run: EXIT 0 · 463 assertions · 0 failures.
Standing advances from eighty-eight (#88 WOW II 427) to eighty-nine. 399a ≠ 399c.
Sources
- graffiti-verification README §7bq
- verify/verify_conj399a.py
- WOW II
- DeLaViña & Pepper 2012 (DP12) on α₂ ≤ WP(Ḡ)+1