A multipartite version of the Hajnal-Szemer\'edi theorem for graphs and hypergraphs

TL;DR

This paper extends the Hajnal-Szemerédi theorem to multipartite graphs and hypergraphs, establishing minimum degree conditions for perfect matchings and confirming a conjecture asymptotically.

Contribution

It provides a multipartite and hypergraph generalization of the Hajnal-Szemerédi theorem, verifying a conjecture for large graphs.

Findings

Minimum degree condition ensures perfect $K_t$-matching in multipartite graphs.

Asymptotic verification of Fisher's conjecture.

Extension to hypergraphs with codegree conditions.

Abstract

A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemer\'edi states that if $G$ is a graph of order $n$ with minimum degree $\delta(G) \ge (t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a $t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t + \gamma)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect $K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore, we consider a generalisation to hypergraphs in terms of the codegree.