Asymptotic multipartite version of the Alon–Yuster theorem

Abstract

In this paper, we prove the asymptotic multipartite version of the Alon–Yuster theorem, which is a generalization of the Hajnal–Szemerédi theorem: If k≥3 is an integer, H is a k-colorable graph and γ>0 is fixed, then, for every sufficiently large n, where |V(H)| divides n, and for every balanced k-partite graph G on kn vertices with each of its corresponding (k2) bipartite subgraphs having minimum degree at least (k−1)n/k+γn, G has a subgraph consisting of kn/|V(H)| vertex-disjoint copies of H.The proof uses the Regularity method together with linear programming.