Counterexample to Written on the Wall II Conjecture 209

Written on the Wall II Conjecture 209 is false.

The conjecture states: if G is a simple connected graph with n > 1 such that (1/6)·[1 + 2·|E(Ḡ)|] ≤ frequency of λ_max(G), then G has a Hamiltonian path. (Note the complement Ḡ: the overline is a CSS text decoration on the WOWII page and silently disappears in plain-text copies.)

Counterexample family: for every m ≥ 7, take K_m with a distinguished vertex s; add two nonadjacent vertices, each adjacent to all of the clique except s; and attach two pendant vertices to s. Then λ_max = 3 is attained exactly at the m clique vertices and |E(Ḡ)| = 2m + 6, so the premise reads 4m + 13 ≤ 6m, which holds for all m ≥ 7. Both pendant vertices must be endpoints of any Hamiltonian path, yet both attach to the same vertex s — the second and penultimate vertices of the path would coincide — so no Hamiltonian path exists.

The smallest member (m = 7) has 11 vertices, graph6 `J?o}]^Nr}^_`. An exhaustive search over all 1,018,690,325 connected graphs on 4 ≤ n ≤ 11 vertices shows it is the unique counterexample of minimum order.

A Lean certificate verifies the statement's negation; a PR will link it.

Start with issue #4507 and the Lean certificate mentioned in its body, using the stated m = 7 graph family as the counterexample. Confirm that the formal work verifies the conjecture's negation and that the smallest example is represented as described; done means the repository contains a reviewed certificate for the false statement.

Written by the indexing model from the issue text.