English

Counting tight Hamilton cycles in Dirac hypergraphs

Combinatorics 2026-04-17 v1

Abstract

Suppose GG is a kk-uniform hypergraph on nn vertices such that every (k1)(k-1)-subset SS of V(G)V(G) belongs to at least δn\delta n edges, where δ>1/2\delta> 1/2. Let Ψ(G)\Psi(G) denote the number of tight Hamilton cycles in GG, that is, cyclic orderings of V(G)V(G) in which every kk consecutive vertices form an edge. We prove that logΨ(G)kh(G)nlog(nk1)+nlognnlogeo(n)\log\Psi(G)\ge kh(G)-n\log{n\choose k-1}+n\log n-n\log e-o(n), where h(G)h(G) is the hypergraph entropy of GG, defined via perfect fractional matchings. This bound is tight, for example, for all (nearly) regular hypergraphs, in particular for the binomial random hypergraph. It also implies a conjecture by Ferber, Hardiman and Mond, stating that Ψ(G)(δo(1))nn!\Psi(G)\ge (\delta-o(1))^n n!.

Keywords

Cite

@article{arxiv.2604.14978,
  title  = {Counting tight Hamilton cycles in Dirac hypergraphs},
  author = {Felix Joos and Xinyue Xie},
  journal= {arXiv preprint arXiv:2604.14978},
  year   = {2026}
}

Comments

23 pages + 9 pages appendix for general l-cycles

R2 v1 2026-07-01T12:12:36.557Z