On multipartite Hajnal–Szemerédi theorems

Let GG be a kk-partite graph with nn vertices in parts such that each vertex is adjacent to at least δ∗(G)δ∗(G) vertices in each of the other parts. Magyar and Martin (2002) [18] proved that for k=3k=3, if δ∗(G)≥23n+1 and nn is sufficiently large, then GG contains a K3K3-factor (a spanning subgraph consisting of nn vertex-disjoint copies of K3K3). Martin and Szemerédi (2008) [19] proved that GG contains a K4K4-factor when δ∗(G)≥34n and nn 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≥3k≥3 and may be utilized to prove a general and tight multipartite Hajnal–Szemerédi theorem.