Counting Perfect Matchings in Dirac Hypergraphs

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 Sárközy, Selkow and Szemerédi 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 $$d d < k , and for n divisible by k , let $$m_{d}(k,n)$$ m d ( k , n ) be the minimum d -degree that ensures the existence of a perfect matching in an n -vertex k -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of $$m_{d}(k,n)$$ m d ( k , n ) , but we are nonetheless able to prove an analogue of the Cuckler–Kahn theorem, showing that if an n -vertex k -uniform hypergraph G has minimum d -degree at least $$(1+\gamma )m_{d}(k,n)$$ ( 1 + γ ) m d ( k , n ) (for any constant $$\gamma >0$$ γ > 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–Kühn–Osthus–Pfenninger and Pham–Sah–Sawhney–Simkin.