Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs

TL;DR

This paper establishes entropy-based bounds on perfect matchings in bipartite hypergraphs and applies these results to Latin squares and hypergraph colorings, advancing combinatorial enumeration techniques.

Contribution

It introduces new upper bounds on the number of perfect matchings and proper colorings in hypergraphs with small maximum codegree, with applications to Latin squares.

Findings

Bound on the number of $A$-perfect matchings in hypergraphs.

Existence of Latin squares with limited transversals.

Upper bound on proper $q$-edge-colorings in regular hypergraphs.

Abstract

A hypergraph is \textit{bipartite with bipartition $(A, B)$} if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is \textit{$A$-perfect} if it saturates every vertex in $A$. We prove an upper bound on the number of $A$-perfect matchings in uniform hypergraphs with small maximum codegree. Using this result, we prove that there exist order-$n$ Latin squares with at most $(n/e^{2.117})^n$ transversals when $n$ is odd and $n \equiv 0\pmod 3$. We also show that $k$-uniform $D$-regular hypergraphs on $n$ vertices have at most $((1+o(1))q/e^k)^{Dn/k}$ proper $q$-edge-colorings when $q = (1+o(1))D$ and the maximum codegree is $o(q)$.