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Claim. Every graph with minimum degree at least $3$ on at most $23$ vertices contains a cycle of length $4$ or $8$, so a counterexample to the conjecture of Problem 64 has at least $24$ vertices, improving the published bound of $16$. The result is Daniel Garcia, Small graphs without power-of-two cycles: a lower bound of 24, a correction to a construction of Exoo, and explicit bounds for f(k), arXiv:2609.04686, posted 2026-09-04 (the claim's date; nine pages), whose abstract describes the proof as a SAT-based exhaustive search certified by DRAT proofs, with the data, code and certificates deposited on Zenodo at the record linked above. The abstract adds that the smallest graph of minimum degree $3$ with no $4$-cycle and no $8$-cycle has exactly $24$ vertices, which does not make it a counterexample, since longer powers of two are not excluded; the paper also repairs a lemma in Exoo's $450$-vertex construction for the bound $f(5)\le450$, which is outside this problem's question. Read depth (2026-10-07): the arXiv record and abstract; the proof was not read and the certificates were not replayed. The thread's family list of 6 December 2025 predates the paper, and the site's commentary does not credit it.

Covers. The statement of Problem 64 for graphs on at most $23$ vertices, where the cycle found has length $2^2$ or $2^3$; Markström's search (claim page) covers cubic graphs on at most $29$ vertices, which this result does not reach.

Depends on. No page of this wiki.

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