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

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 $V_{۱}‎, ‎V_{۲}‎,‎\ldots‎,‎V_{k}$ such that the subhypergraph generated on $\bigcup_{i=۱}^{k-۱}V_{i}$ has a unique perfect matching‎. ‎In this case‎, ‎we give a necessary and sufficient condition for having a matching of size $t=|V_{۱}|$ in $\mathcal{H}$‎. ‎Some relevant results and counterexamples are given as well‎.

Keywords:

$k$-uniform $k$-partite hypergraph‎ , ‎matching‎ , ‎perfect matching‎ , ‎vertex cover‎ , ‎Hall&#۰۳۹ , s theorem