Minimum co-degree condition for perfect matchings in k-partite k-graphs

Abstract

Let $H$ be a $k$-partite $k$-graph with $n$ vertices in each partition class, and let ${\delta }_{k-1}(H)$ denote the minimum co-degree of $H$. We characterize those $H$ with ${\delta }_{k-1}(H)\ge n/2$ and with no perfect matching. As a consequence we give an affirmative answer to the following question of Rödl and Ruciński: If $k$ is even or $n\not\equiv 2(\mathrm{mod}4)$, does ${\delta }_{k-1}(H)\ge n/2$ imply that $H$ has a perfect matching? We also give an example indicating that it is not sufficient to impose this degree bound on only two types of $(k-1)$-sets.