Almost Perfect Matchings in $k$-Partite $k$-Graphs

Abstract

The minimum co-degree threshold for a perfect matching in a k-graph with n vertices was determined by Rödl, Ruciński and Szemerédi for the case when n ≡ 0 (mod k). Recently, Han resolved the remaining cases when n ≡ 0 (mod k), establishing a conjecture of Rödl, Ruciński and Szemerédi. In this paper, we determine the minimum co-degree threshold for almost perfect matchings in k-partite k-graphs, answering a question of Rödl and Ruciński.

Matchings in k‐partite k‐uniform hypergraphs

“…It was announced at [28] and appeared in the dissertation of the second author [27]. The second case of Corollary 1.2 resolves [ [20], Problem 3.14] and was independently proven by Lu et al [17].…”

“…is odd. When proving Corollary 1.2 directly, the authors of [17,27] closely followed the approach used by the first author [6] by separating two cases based on whether H is close to H 0 . In contrast, to prove Theorem 1.1, we have to consider three cases separately: when H is close to H 0 , when H is close to (a weaker form of) H 1 , and when H is far from both H 0 and H 1 .…”

Minimum codegree condition for perfect matchings in k‐partite k‐graphs

“…In Section 2, we deal with the case when H is ε-close to H k n ( , ) 0 for some sufficiently small ε. In Section 3, we deal with the case when H is not ε-close to H k n ( , ) 0 , using the absorbing method from [12] and a recent result of the authors [9] (see Lemma 3.1).…”

“…Rödl and Ruciński [11] asked the following question: is it true that δ H n k ( ) k−1 ≥ ∕ implies that H has a matching of size at least n − 1? The present authors [9] and, independently, Han, Zang, and Zhao [5] such that the following holds. Let H be a k-partite k-graph with n n 1 ≥ vertices in each partition class and with δ H n n ( ) (1 2 − 1 log )…”

“…Kühn and Osthus [6] showed that the minimum codegree threshold for the existence of a perfect matching in a k-partite k-graph with n vertices in each partition class is between n 2 ∕ and n n n 2 + 2 log ∕ . Lu, Wang, and Yu [9] and, independently, Han, Zang, and Zhao [5] showed that n k ∕ is the minimum codegree threshold for a k-partite kgraph H with n vertices in each partition class to admit a matching of size V H k ( ) − | | . Aharoni, Georgakopoulos, and Sprüssel [1] obtained the following result which is stronger than the result of Kühn…”