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.
Abstract. Paul Seymour conjectured that any graph G of order n and minimum degree of at least k-~T1 n contains the kth power of a Hamiltonian cycle. Here, we prove this conjecture for sufficiently large n.
We prove -for sufficiently large n -the following conjecture of Faudree and Schelp:2n − 1 for odd n, 2n − 2 for even n, for the three-color Ramsey numbers of paths on n vertices.
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