Exact minimum codegree threshold for $K_4^{-}$-factors

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.

23 pages

References in corpus (1)

• Minimum vertex degree thresholds for tiling complete 3-partite 3-graphs

Cited by in corpus (2)

• Triangle-degrees in graphs and tetrahedron coverings in 3-graphs
• Codegree thresholds for covering 3-uniform hypergraphs