Proof of the Alon–Yuster conjecture Original Research Article

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 h1⩽h2⩽⋯⩽hk, then the conjecture is true with c(H)=hk+hk−1−1.