Matching stability for 3-partite 3-uniform hypergraphs

TL;DR

This paper establishes a stability condition for 3-partite 3-uniform hypergraphs, linking edge count and the size of maximum matchings and vertex covers, with a tight bound for large n.

Contribution

It provides a new stability theorem for 3-partite 3-uniform hypergraphs relating edge density to matchings and vertex covers, extending prior results.

Findings

If e(G) ≥ (s-1)n^2 + 3n - s, then G has a vertex cover of size s.

The bound on edges is tight for large n.

The result applies to hypergraphs with at least 162 vertices per class.

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.