---

Claim. Every simple cubic bipartite graph on at most $58$ vertices contains a cycle of length $4$, $8$ or $16$, so a cubic bipartite counterexample to the conjecture of Problem 64 has at least $60$ vertices, a cubic bipartite graph having even order; the abstract says this improves the published bound for the class from $30$ to $60$. The result is Julius Tranquilli, A 60-Vertex Lower Bound for Cubic Bipartite Counterexamples to the Erdős-Gyárfás Conjecture, arXiv:2608.02675, posted 2026-08-02 (the claim's date; nineteen pages). The abstract describes the proof: below $62$ vertices a cubic bipartite graph without $4$- and $8$-cycles contains a $6$-cycle by a Moore-bound observation; the graph is the Levi graph of a linear symmetric $v_3$-configuration, in which that $6$-cycle is a Berge triangle with, up to symmetry, two rooted extensions; a restricted-growth search on at most $29$ points exhausts both trees, and the computation is checked by two separately implemented exact procedures and a static witness certificate, with source code and certificates in an accompanying repository and Zenodo archive. Read depth: the arXiv record and abstract; the proof was not read and the certificates were not replayed. A thread post of 31 August 2026 cites the paper by title and number and reports an exhaustive search of its own reaching $62$ vertices, with AI assistance as the poster discloses; the post is not a dated manuscript and gets no page.

Covers. The statement of Problem 64 for cubic bipartite graphs on at most $58$ vertices, where the cycle found has length $2^2$, $2^3$ or $2^4$; cubic graphs that are not bipartite are covered only up to $29$ vertices, by Markström's search (claim page), and bipartite graphs with a vertex of degree above $3$ only up to $31$ vertices, by Salehi Nowbandegani and Esfandiari (claim page).

Depends on. No page of this wiki.

Standing. Claimed: an arXiv preprint with no refereed version or outside review known to this corpus; the computation is author-reported and has not been replayed here. The site labels the problem FALSIFIABLE and its commentary does not credit the paper.