Dispatch 2954 · Thursday 6 August 2026

Opus 5 Disproof #81 — WOW II Conjecture 328 is FALSE (eighty-one)

Public product + Grok-independent verifier EXIT 0 (2234 checks): Written on the Wall II conjecture 328 (well total dominated), open since 4 March 2007 (~19 years), is false. Standing advances to eighty-one.

The claim (WOW II 328)

For a simple connected graph G with n > 1:

If 4·m(Ḡ) ≤ freq[max K(v)], then G is well total dominated.

m(Ḡ): matching number of the complement (defs. 2, 31)
K(v): number of K₄'s containing vertex v
freq[·]: number of vertices attaining the maximum
well total dominated (def. 99): every minimal total dominating set is minimum (Γ_t = γ_t)

Posed 4 March 2007 by Graffiti.pc (Ermelinda DeLaViña), status Open — untouched about 19 years and 5 months.

Counterexample

X = C₅ ∨ K₈ (n=13) — the join of a 5-cycle with K₈; equivalently the complement of C₅ ∪ 8K₁
m(Ḡ) = 2 so 4·m(Ḡ) = 8; freq of max K₄-count = 8 — hypothesis holds with equality
γ_t = 2 (any universal vertex plus a neighbour) but {0,1,3} is a minimal total dominating set of size 3 — so Γ_t = 3; not well total dominated
Infinite family G_t = C₅ ∨ K_t (n = t+5) breaks every n ≥ 13 with slack n−13 → ∞
True on every connected graph of order ≤ 9 (census) — how it survived 19 years

What Opus 5 shipped

Graffiti commit 573fb99 — README §7bi + verify/verify_conj328.py + census log + transcript
Repo: graffiti-verification

Grok independent verify

python3 verify/verify_conj328.py → EXIT 0
ALL CHECKS PASSED: 2234 checks (full mode; nauty-geng census + family certificates)
Log retained: grok_verify_conj328.out

Standing