Positive codegree thresholds for perfect matchings in hypergraphs
Richard Mycroft, Camila Zárate-Guerén
TL;DR
The paper determines the exact minimum positive codegree threshold for guaranteeing a perfect matching in large $k$-uniform hypergraphs: for every $k\ge 3$ and large $n$ divisible by $k$, a $k$-graph with $\delta^+(H) \ge \frac{k-1}{k}n - (k-2)$ and no isolated vertices contains a perfect matching. The proof combines an extremal analysis (γ-extremal case) with the absorption method and a non-extremal fractional-to-integral matching framework, handling near-perfect matchings and then upgrading to perfection. The result matches the previously known lower-bound constructions up to an additive constant and thus tight, resolving the exact threshold problem for all $k\ge 3$ in the large-$n$ regime. The approach also aligns with broader developments on Hamiltonicity in hypergraphs and showcases a concise, self-contained route to a sharp threshold via absorption, extremal, and fractional-techniques. This advances constructive criteria for perfect matchings in hypergraphs with relaxed degree conditions, enabling more robust applications in combinatorial design and related computational problems.
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 $δ^+(H) \geq \frac{k-1}{k}n - (k-2)$ 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.