A generalization of hall's theorem for k-uniform k-partite hypergraphs

In this paper we prove a generalized version of Hall's theorem for hypergraphs. More precisely, let H be a k-uniform k- partite hypergraph with some ordering on parts as V1, V2,..., Vk. such that the subhypergraph generated on union of V1, V2,..., Vk-1 has a unique perfect matching. In this case, we give a necessary and sufficient condition for having a matching of size t = |V1| in H. Some relevant results and counterexamples are given as well.