Counting tight Hamilton cycles in Dirac hypergraphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908970395893760 |
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| author | Joos, Felix Xie, Xinyue |
| author_facet | Joos, Felix Xie, Xinyue |
| contents | Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $δn$ edges, where $δ> 1/2$. Let $Ψ(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic orderings of $V(G)$ in which every $k$ consecutive vertices form an edge. We prove that $\logΨ(G)\ge kh(G)-n\log{n\choose k-1}+n\log n-n\log e-o(n)$, where $h(G)$ is the hypergraph entropy of $G$, 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 $Ψ(G)\ge (δ-o(1))^n n!$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting tight Hamilton cycles in Dirac hypergraphs Joos, Felix Xie, Xinyue Combinatorics Suppose $G$ is a $k$-uniform hypergraph on $n$ vertices such that every $(k-1)$-subset $S$ of $V(G)$ belongs to at least $δn$ edges, where $δ> 1/2$. Let $Ψ(G)$ denote the number of tight Hamilton cycles in $G$, that is, cyclic orderings of $V(G)$ in which every $k$ consecutive vertices form an edge. We prove that $\logΨ(G)\ge kh(G)-n\log{n\choose k-1}+n\log n-n\log e-o(n)$, where $h(G)$ is the hypergraph entropy of $G$, 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 $Ψ(G)\ge (δ-o(1))^n n!$. |
| title | Counting tight Hamilton cycles in Dirac hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.14978 |