Perfect matchings in r-partite r-graphs

TL;DR

This paper proves that an r-partite r-graph with certain degree conditions on two sides must contain a perfect matching, answering a question posed by Kuhn and Osthus.

Contribution

It establishes a new sufficient condition for the existence of perfect matchings in r-partite r-graphs, advancing understanding in combinatorial graph theory.

Findings

Proves the existence of perfect matchings under specified degree conditions.

Answers an open question by Kuhn and Osthus.

Provides a new criterion for perfect matchings in r-partite r-graphs.

Abstract

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H, each satisfying the following condition: the degree of each legal (r-1)-tuple contained in the complement of this side is strictly larger than n/2. We prove that under this condition H must have a perfect matching. This answers a question of Kuhn and Osthus.