A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs

TL;DR

This paper improves the minimum degree threshold for perfect matchings in 3-uniform hypergraphs by introducing degree sequence conditions, allowing a third of vertices to have slightly lower degrees, and demonstrates the near-optimality of these results.

Contribution

It introduces a new family of degree sequence conditions that strengthen the vertex degree threshold for perfect matchings in 3-uniform hypergraphs, extending previous results.

Findings

Improved threshold conditions for perfect matchings.

Allows a third of vertices to have degrees below the threshold.

Shows the tightness of the new degree sequence results.

Abstract

The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of H\`{a}n, Person and Schacht who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an $n$-vertex $3$-graph is $\left(\frac{5}{9}+o(1)\right)\binom{n}{2}$. In this paper we improve on this result, giving a family of degree sequence results, all of which imply the result of H\`{a}n, Person and Schacht, and additionally allow one third of the vertices to have degree $\frac{1}{9}\binom{n}{2}$ below this threshold. Furthermore, we show that this result is, in some sense, tight.