Vertex degree sums for perfect matchings in 3-uniform hypergraphs

TL;DR

This paper establishes a tight minimum degree sum condition for pairs of vertices in large 3-uniform hypergraphs to guarantee the existence of a perfect matching, advancing understanding in hypergraph matching theory.

Contribution

It determines a precise, tight degree sum threshold ensuring perfect matchings in large 3-uniform hypergraphs without isolated vertices.

Findings

Derived a tight bound for degree sums guaranteeing perfect matchings.

Proved the bound is optimal through extremal examples.

Extended hypergraph matching theory with new degree sum conditions.

Abstract

We determine the minimum degree sum of two adjacent vertices that ensures a perfect matching in a 3-graph without isolated vertex. More precisely, suppose that $H$ is a 3-uniform hypergraph whose order $n$ is sufficiently large and divisible by $3$. If $H$ contains no isolated vertex and $deg(u)+ deg(v) > \frac{2}{3}n^2-\frac{8}{3}n+2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching. This bound is tight.