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 V-1, ..., V-k. 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 boolean OR g is not an element of 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 2n. If for all the legal 2-tuple f with | f boolean AND V-1 | = 1 and for all the legal 2-tuple g with |g boolean AND V-1| = 0, we have d(f) >= n - 2 and d(g) > n/2, then G has a perfect matching.