Proof of the Alon-Yuster conjecture

In this paper we prove the following conjecture of Alon and Yuster. Let H be a graph with h vertices and chromatic number k. There exist constants c(H) and n0(H) such that if n¿n0(H) and G is a graph with hn vertices and minimum degree at least (1 − 1=k)hn + c(H), then G contains an H-factor. In fact, we show that if H has a k-coloring with color-class sizes h16h26 · · · 6h k , then the conjecture is true with c(H)=h k +h k−1 −1.