Quadripartite version of the Hajnal-Szemerédi theorem (Q941371)

The paper deals with quadripartite graphs. In the first section, the quadripartite version of the Hajnal-Szemeredi theorem is presented. The proof of this theorem is similar to the proof of the tripartite version of the Hajnal-Szemeredi theorem, which was previously published by the second author of this paper. In the second section, it is proven that a balanced quadripartite graph of \(4N\) vertices has a perfect K4-factor, even if the minimum degree is a little bit less than \(3/4N\). In the third section, it is proven that a balanced quadripartite graph of \(4N\) vertices such that each vertex is adjacent to at least \(3/4N\) vertices in each of other classes, has a perfect K4-factor for \(N\) large enough. The case of \(N/4\) not being an integer is studied in the fourth section. It is proven that in this case the quadripartite graph has also a K4-factor.