---

Claim. There is an absolute constant $c>0$ such that every $(n+1)$-regular graph $G$ on $2n$ vertices has at least $c\,2^{2n}$ vertex subsets whose induced graph has a Hamilton cycle, and the count is in fact at least $(\tfrac12-o(1))2^{2n}$. These are Theorem 2.2 and Theorem 4.1 of N. Draganić, P. Keevash and A. Müyesser, Cyclic subsets in regular Dirac graphs, Int. Math. Res. Not. IMRN 2025, no. 14, rnaf215, 1--16, first posted as arXiv:2503.01826 on 2025-03-03 (this page's date). The first theorem answers the question of Problem 622 affirmatively with no further hypothesis, so the claim is full; the second raises the constant to $\tfrac12-o(1)$, which is asymptotically best because the paper's Lemma 5.1 computes the proportion $\tfrac12+O(n^{-1/2})$ for $K_{n-1,n+1}$ with a $2$-factor added inside its larger part. The proof classifies $(n+1)$-regular graphs on $2n$ vertices into bidense graphs, two almost-cliques and almost-bipartite graphs, and in the last case uses linear forests inside the parts to correct the imbalance of a random subset. The corpus states the two theorems and rewrites their proofs on the Theorem 2.2 and Theorem 4.1 result pages of its source card. The paper's exact result, Theorem 1.2, which names a minimizing family for large $n$, is not part of this claim; the card records reconstruction gaps in its finer proof.

Acceptance. Refereed: International Mathematics Research Notices is a refereed journal, and the paper is its version of record, received 19 March 2025, accepted 27 June 2025 and published online 22 July 2025. Reviewed: T. F. Bloom, the site's curator, who took no part in the paper, labels the problem PROVED, credits the resolution to this paper and states its asymptotic bound in the commentary (snapshot of 2026-09-05; the one comment in the thread concerns a broken reference link, and the proof-claims tab is empty). The corpus's rewritten proofs on the result pages are author-recorded compilation work, not an independent review, and warrant no evidence kind. No second paper attesting the theorem is held.

Depends on. Nothing in this wiki: the proof is the paper's.