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Claim. Every graph with minimum degree at least $3$ and no induced path on thirteen vertices contains a cycle whose length is a power of $2$, so the conjecture of Problem 64 holds for $P_{13}$-free graphs, which include the $P_{10}$-free graphs of Hu and Shen (claim page) and the $P_8$-free graphs of Gao and Shan (claim page). The result is Anand Shripad Hegde, R. B. Sandeep and P. Shashank, Erdős-Gyárfás conjecture on graphs without long induced paths, arXiv:2410.22842, posted 2024-10-30 (the claim's date; v2 of 2025-02-11, six pages), whose abstract says the proof is obtained with the aid of a computer search. Read depth: the arXiv record and abstract; the proof was not read. Duran Ballester's manuscript (claim page) cites the theorem; the site's remark and the thread's family list of 6 December 2025 do not name it.

Covers. The statement of Problem 64 for $P_{13}$-free graphs.

Depends on. No page of this wiki.

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