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

TL;DR

This paper characterizes conditions under which k-partite k-graphs with high minimum co-degree lack perfect matchings, providing an answer to a question by R"odl and Ruciński and exploring degree conditions for perfect matchings.

Contribution

It offers a complete characterization of k-partite k-graphs with minimum co-degree at least n/2 that do not have perfect matchings, addressing an open problem in hypergraph theory.

Findings

Characterization of k-partite k-graphs with no perfect matching at δ_{k-1}(H) ≥ n/2.

Affirmative answer to R"odl and Ruciński's question for even k or n ≢ 2 mod 4.

Example showing degree conditions on only two types of (k-1)-sets are insufficient.

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 also give an example indicating that it is not sufficient to impose this degree bound on only two types of $(k-1)$-sets.