Proof of the Alon–Yuster conjecture

Abstract

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 n 0( H) such that if n⩾ n 0( 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 h 1⩽ h 2⩽⋯⩽ h k , then the conjecture is true with c( H)= h k + h k−1 −1.