Matching stability for 3-partite 3-uniform hypergraphs

Abstract

Let $n,k,s$ be three integers such that $k\geq 2$ and $n\geq s\geq 1$. Let $H$ be a $k$-partite $k$-uniform hypergraph with $n$ vertices in each class. Aharoni (2017) showed that if $e(H)>(s-1)n^{k-1}$, then $H$ has a matching of size $s$. In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if $G$ is a $3$-partite $3$-uniform hypergraph with $n\geq 162$ vertices in each class, $e(G)\geq (s-1)n^2+3n-s$ and $G$ contains no matching of size $s+1$, then $G$ has a vertex cover of size $s$. Our bound is also tight.

Theorems & Definitions (13)

• Theorem 1.1: Berge, Berge75
• Theorem 1.2: Aharoni and Howard, AH17
• Theorem 1.3
• Conjecture 1.4
• Lemma 2.1: Lo and Markström, LM14
• Theorem 2.2: Aharoni and Howard, AH17
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6