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arXiv · 1509.02577

Exact minimum codegree threshold for $K^- _4$-factors

Jie Han·Allan Lo·Andrew Treglown·Yi Zhao
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Abstract

Given hypergraphs $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ which cover all the vertices in $H$. Let $K^- _4$ denote the $3$-uniform hypergraph with $4$ vertices and $3$ edges. We show that for sufficiently large $n\in 4 \mathbb N$, every $3$-uniform hypergraph $H$ on $n$ vertices with minimum codegree at least $n/2-1$ contains a $K^- _4$-factor. Our bound on the minimum codegree here is best-possible. It resolves a conjecture of Lo and Markström for large hypergraphs, who earlier proved an asymptotically exact version of this result. Our proof makes use of the absorbing method as well as a result of Keevash and Mycroft concerning almost perfect matchings in hypergraphs.

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Jie Han, Allan Lo, Andrew Treglown, Yi Zhao. 2015-09-08. Exact minimum codegree threshold for $K^- _4$-factors. https://arxiv.org/abs/1509.02577

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