On Perfect Matchings in Uniform Hypergraphs with Large Minimum Vertex Degree

Abstract

We study sufficient $\ell$-degree ($1\leq\ell<k$) conditions for the appearance of perfect and nearly perfect matchings in k-uniform hypergraphs. In particular, we obtain a minimum vertex degree condition ($\ell=1$) for 3-uniform hypergraphs, which is approximately tight, by showing that every 3-uniform hypergraph on n vertices with minimum vertex degree at least $(5/9+o(1))\binom{n}{2}$ contains a perfect matching.