Entropy Bounds for Perfect Matchings in Bipartite 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 $n\equiv 0(\mathrm{mod}3)$. 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 $o(q)$.

10 pages, 1 figure; published in the Electronic Journal of Combinatorics