Average Eccentricity, Induced Bipartite Subgraphs, and Hamiltonian Paths

For a finite simple connected graph G, let b(G) be the maximum order of an induced bipartite subgraph, and let avec(G) be the average of the vertex eccentricities. Written on the Wall II, Conjecture 198a asserts that b(G) ≤ 2 + avec(G) implies that G has a Hamiltonian path.

This preprint presents a candidate proof of the conjecture. Its main ingredient is a geodesic lemma showing that every connected graph with no Hamiltonian path satisfies b(G) ≥ diam(G) + 2. Combining this lemma with a known radius bound and the Chvátal–Erdős path theorem reduces the conjecture to an elementary contradiction in diameter two.

The accompanying reproducibility archive contains the LaTeX source, exact-search programs, computational logs, and SHA-256 checksums. The computational checks are supporting evidence and are not used in the proof.

Version 0.1. This manuscript has not been peer reviewed. The investigation and manuscript preparation were substantially assisted by artificial-intelligence systems, as detailed in the paper. Readers should verify every claim independently.

Update, September 2026: This result has now been formally verified in Lean 4. The complete formalization is available on GitHub, where GitHub Actions recompiles the Aristotle-generated project with Lean, checks the displayed proof against the verified source, and audits its axiom dependencies.

• Permanent Lean proof: https://github.com/bzcohen588/written-on-the-wall-198a/blob/5481927df49c374b06fa0931b5aa24f9296faa04/Conjecture198a.lean
• Successful verification run: https://github.com/bzcohen588/written-on-the-wall-198a/actions/runs/35154803656