Positive codegree thresholds for perfect matchings in hypergraphs

TL;DR

This paper determines the exact minimum positive codegree condition needed for large hypergraphs to contain perfect matchings, extending known results to all uniformities $k \\geq 3$ with tight bounds.

Contribution

It establishes the precise best possible minimum positive codegree thresholds for perfect matchings in large $k$-uniform hypergraphs, generalizing and refining previous bounds.

Findings

Exact codegree thresholds for $k=3$ and $k \\geq 4$.

Thresholds are tight up to an additive constant.

No isolated vertices are required for the existence of perfect matchings.

Abstract

We give, for each $k \geq 3$, the precise best possible minimum positive codegree condition for a perfect matching in a large $k$-uniform hypergraph $H$ on $n$ vertices. Specifically we show that, if $n$ is sufficiently large and divisible by $k$, and $H$ has minimum positive codegree $\delta^+(H) \geq \frac{k-1}{k}n - (k-2)$ and no isolated vertices, then $H$ contains a perfect matching. For $k=3$ this was previously established by Halfpap and Magnan, who also gave bounds for $k \geq 4$ which were tight up to an additive constant.