Co-degree condition for 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 $\delta_{k-1}(H) \geq n/2$ and with no perfect matching. As a consequence we give an affirmative answer to the following question of R\"odl and Ruci\'nski: If $k$ is even or $n \not\equiv 2 \pmod 4$, does $\delta_{k-1}(H) \geq n/2$ imply that $H$ has a perfect matching? We give an example indicating that it is not sufficient to impose this degree bound on only two types of $(k-1)$-sets. For near perfect matching, we gave a tight sufficient condition in term of co-degree, which is also independently obtained by Han, Zang and Zhao. Moreover, I would like to introduce several problems I am interested in.

combinatorics

Audience: researchers in the topic

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SCMS Combinatorics Seminar

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