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arXiv · 2408.09589

Counting Perfect Matchings In Dirac Hypergraphs

Matthew Kwan·Roodabeh Safavi·Yiting Wang
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Abstract

One of the foundational theorems of extremal graph theory is Dirac's theorem, which says that if an n-vertex graph G has minimum degree at least n/2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by S\'arkozy, Selkow and Szemer\'edi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler and Kahn's result to perfect matchings in hypergraphs. For positive integers d < k, and for n divisible by k, let $m_{d}(k,n)$ be the minimum d-degree that ensures the existence of a perfect matching in an n-vertex k-uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of $m_{d}(k,n)$, but we are nonetheless able to prove an analogue of the Cuckler-Kahn theorem, showing that if an n-vertex k-uniform hypergraph G has minimum d-degree at least $(1+\gamma)m_{d}(k,n)$ (for any constant $\gamma>0$), then the number of perfect matchings in G is controlled by an entropy-like parameter of G. This strengthens cruder estimates arising from work of Kang-Kelly-K\"uhn-Osthus-Pfenninger and Pham-Sah-Sawhney-Simkin.

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Matthew Kwan, Roodabeh Safavi, Yiting Wang. 2024-08-18. Counting Perfect Matchings In Dirac Hypergraphs. https://arxiv.org/abs/2408.09589

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