Counting Perfect Matchings In Dirac Hypergraphs

TL;DR

This paper advances the program of counting spanning structures by extending the CK entropy framework from graphs to hypergraphs. It proves a hypergraph analogue: for a $k$-uniform hypergraph on $n$ vertices with minimum $d$-degree exceeding the Dirac threshold by a constant factor, the number of perfect matchings is controlled by a graph-entropy parameter $h(G)$, via $\ ext{e}^{h(G)}(1/ ext{e}+o(1))^{(1-1/k)n}$. The authors develop an annealing technique to handle maximum-entropy fractional perfect matchings in hypergraphs, pair this with a random greedy process, and establish both a lower bound (through the process) and an upper bound (via entropy methods). A specialized bound for $d\ge k/2$ yields a more explicit lower bound in terms of $p=\delta_d(G)/\binom{n-d}{k-d}$ and the complete $k$-graph PM count, strengthening previous partial results. Together, these results generalize CK’s entropy approach to the hypergraph setting and open avenues for further extensions to other Dirac-type parameters and spanning substructures.

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 Sárkozy, 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 < 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 $(1+γ)m_{d}(k,n)$ (for any constant $γ>0$), 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.

Figures (1)

• Figure 1: We illustrate a shifting structure for $k=3$. The three dotted edges, from top to bottom, are $f_1(= f)$, $f_2$ and $f_3$, and the three solid edges, from left to right, are $e_1 (= e)$, $e_2$ and $e_3$. The set $U_2$ contains the two square-shaped vertices and the set $U_3$ contains the two triangle-shaped vertices.

Theorems & Definitions (40)

• Definition 1.1: Fractional perfect matchings
• Definition 1.2: Graph entropy
• Theorem 1.3: CK-entropyCK-counting
• Theorem 1.4: CK-entropyCK-counting
• Definition 1.5: Hypergraph degrees
• Definition 1.6: Hypergraph Dirac threshold
• Theorem 1.7
• Theorem 1.8
• Definition 2.1: Dirac hypergraphs
• Definition 2.2