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arXiv · 2506.17652

Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs

Tantan Dai·Alexander Divoux·Tom Kelly
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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 $o(q)$.

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Tantan Dai, Alexander Divoux, Tom Kelly. 2025-06-21. Entropy Bounds for Perfect Matchings in Bipartite Hypergraphs. https://doi.org/10.37236/14339

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