Perfect Matching in k-partite k-graphs and 3-uniform HM-bipartite Hypergraphs

Let H=(V,E)be an n-balanced k-partite k-graph with partition classes V1,...,Vk.Suppose for every legal(k-1)-tuple f contained in V\V1 and for every legal(k-1)-tuple g contained in V\Vk such that f∪g■E(H),we have d(f)+d(g)≥n+1.In this paper,we prove that under this condition H must have a perfect matching.Another result of this paper is about the perfect matching in 3-uniform hm-bipartite hypergraphs.Let G be a 3-uniform hm-bipartite hypergraph with one of whose sides V1 has the size n,the another side V2 has size 2 n.If for all the legal 2-tuple f with|f∩V1|=1 and for all the legal 2-tuple g with|g∩V1|=0,we have d(f)≥n-2 and d(g)>n/2,then G has a perfect matching.