On multipartite Hajnal–Szemerédi theorems

Let G be a k -partite graph with n vertices in parts such that each vertex is adjacent to at least δ ∗ ( G ) vertices in each of the other parts. Magyar and Martin (2002) [18] proved that for k = 3 , if δ ∗ ( G ) ≥ 2 3 n + 1 and n is sufficiently large, then G contains a K 3 -factor (a spanning subgraph consisting of n vertex-disjoint copies of K 3 ). Martin and Szemerédi (2008) [19] proved that G contains a K 4 -factor when δ ∗ ( G ) ≥ 3 4 n and n is sufficiently large. Both results were proved using the Regularity Lemma. In this paper we give a proof of these two results by the absorbing method. Our absorbing lemma actually works for all k ≥ 3 and may be utilized to prove a general and tight multipartite Hajnal–Szemerédi theorem.