Vertex Degree Sums for Matchings in 3-Uniform Hypergraphs

Abstract

Let n, s be positive integers such that n is sufficiently large and s ≤ n/3. Suppose H is a 3-uniform hypergraph of order n. If H contains no isolated vertex and deg(u) + deg(v) > 2(s − 1)(n − 1) for any two vertices u and v that are contained in some edge of H, then H contains a matching of size s. This degree sum condition is best possible and confirms a conjecture of the authors [Electron. J. Combin. 25 (3), 2018], who proved the case when s = n/3. n−1 2 − n−s 2 and n ≥ 3s, then H contains a matching of size s.