On a tiling conjecture of Komlo's for 3-chromatic graphs

Given two graphs G and H , an H -matching of G (or a tiling of G with H) is a subgraph of G consisting of vertex-disjoint copies of H . For an r-chromatic graph H on h vertices, we write u = u(H) for the smallest possible color-class size in any r-coloring of H . The critical chromatic number of H is the number cr(H)= (r− 1)h=(h− u). A conjecture of Koml# os states that for every graph H , there is a constant K such that if G is any n-vertex graph of minimum degree at least (1− (1= cr(H)))n, then G contains an H -matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all su<ciently large values of n when H is a 3-chromatic graph. c © 2003 Elsevier B.V. All rights reserved.