Perfect matching in 3-uniform hypergraphs with large vertex degree

TL;DR

This paper proves a tight minimum vertex degree condition for the existence of perfect matchings in 3-uniform hypergraphs with 3k vertices, advancing understanding of hypergraph matchings.

Contribution

The authors establish a sharp minimum degree threshold guaranteeing perfect matchings in 3-uniform hypergraphs, supported by a matching construction showing optimality.

Findings

Minimum degree condition for perfect matchings established

Construction demonstrates the bound is tight

Result advances hypergraph matching theory

Abstract

A perfect matching in a 3-uniform hypergraph on $n=3k$ vertices is a subset of $\frac{n}{3}$ disjoint edges. We prove that if $H$ is a 3-uniform hypergraph on $n=3k$ vertices such that every vertex belongs to at least ${n-1\choose 2} - {2n/3\choose 2}+1$ edges then $H$ contains a perfect matching. We give a construction to show that this result is best possible.