d-matching in 3-uniform hypergraphs

A matching in a 3-uniform hypergraph is a set of pairwise disjoint edges. A d-matching in a 3-uniform hypergraph H is a matching of size d. Let V1,V2 be a partition of n vertices such that |V1|=2d−1 and |V2|=n−2d+1. Denote by E3(2d−1,n−2d+1) the 3-uniform hypergraph with vertex set V1∪V2 consisting of all those edges which contain at least two vertices of V1. Let H be a 3-uniform hypergraph of order n≥9d2 such that deg(u)+deg(v)>2[n−12−n−d2] for any two adjacent vertices u,v∈V(H). In this paper, we prove H contains a d-matching if and only if H is not a subgraph of E3(2d−1,n−2d+1).