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arXiv · 2505.17981

Positive codegree thresholds for perfect matchings in hypergraphs

Richard Mycroft·Camila Zárate-Guerén
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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.

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Richard Mycroft, Camila Zárate-Guerén. 2025-05-23. Positive codegree thresholds for perfect matchings in hypergraphs. https://arxiv.org/abs/2505.17981

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