Asymptotic multipartite version of the Alon-Yuster theorem

In this paper, we prove the asymptotic multipartite version of the Alon-Yuster theorem, which is a generalization of the Hajnal-Szemer\'edi theorem: If $k\geq 3$ is an integer, $H$ is a $k$-colorable graph and $\gamma>0$ 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 $\binom{k}{2}$ bipartite subgraphs having minimum degree at least $(k-1)n/k+\gamma n$, $G$ has a subgraph consisting of $kn/|V(H)|$ vertex-disjoint copies of $H$. The proof uses the Regularity method together with linear programming.