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

Abstract

In this paper we prove a generalized version of Hall's theorem in graphs‎, ‎for hypergraphs‎. ‎More precisely‎, ‎let $mathcal{H}$ be a $k$-uniform $k$-partite hypergraph with some ordering on parts as $V_{1}‎, ‎V_{2}‎,‎ldots‎,‎V_{k}$ such that the subhypergraph generated on $bigcup_{i=1}^{k-1}V_{i}$ has a unique perfect matching‎. ‎In this case‎, ‎we give a necessary and sufficient condition for having a matching of size $t=|V_{1}|$ in $mathcal{H}$‎. ‎Some relevant results and counterexamples are given as well‎.