Proof of the Alon-Yuster conjecture (Q5937938)

In this very interesting article, the authors prove the following conjecture of \textit{N. Alon} and \textit{R. Yuster} [J. Comb. Theory, Ser. B 66, No. 2, 269-282 (1996; Zbl 0855.05085)]. Let \(H\) be a graph of order \(h\) and chromatic number \(k\). There exist constants \(c(H)\) and \(n_0(H)\) such that if \(n\geq n_0(H)\) and \(G\) is a graph of order \(hn\) and minimum degree at least \((1-1/k)hn+c(H)\), then \(G\) contains an \(H\)-factor. More precisely, the authors show that if \(H\) has a \(k\)-coloring with color-class sizes \(h_1\leq h_2\leq\ldots\leq h_k\), then the conjecture is true for \(c(H)=h_k+h_{k-1}-1\). Furthermore, an example demonstrates that this is no longer valid with \(c(H)=h_k-2\).