Matching in 3-uniform hypergraphs

Abstract A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. We use E 3 ( 2 d − 1 , n − 2 d + 1 ) to denote the 3-uniform hypergraph whose vertex set can be partitioned into two vertex classes V 1 and V 2 of size 2 d − 1 and n − 2 d + 1 , respectively, and whose edge set consists of all the triples containing at least two vertices of V 1 . Let H be a 3-uniform hypergraph of order n ≥ 13 d with no isolated vertex and deg ( u ) + deg ( v ) > 2 ( n − 1 2 − n − d 2 ) for any two adjacent vertices u , v ∈ V ( H ) . In this paper, we show that H contains a matching of size d if and only if H is not a subgraph of E 3 ( 2 d − 1 , n − 2 d + 1 ) . This result improves our previous one in Zhang and Lu (2018).