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 [20] proved that for k = 3, if δ∗(G) ≥ 2 3 n and n is sufficiently large, then G contains a K3-factor (a spanning subgraph consisting of n vertex-disjoint copies of K3) except that G is one particular graph. Martin and Szemerédi [21] proved that G contains a K4-factor when δ∗(G) ≥ 34n and n is sufficiently large. Both results were proved by 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.