The minimum degree threshold for perfect graph packings

Abstract

Let H be any graph. We determine (up to an additive constant) the minimum degree of a graph G which ensures that G has a perfect H-packing (also called an H-factor). More precisely, let delta(H,n) denote the smallest integer t such that every graph G whose order n is divisible by |H| and with delta(G) > t contains a perfect H-packing. We show that delta(H,n) = (1-1/\chi*(H))n+O(1). The value of chi*(H) depends on the relative sizes of the colour classes in the optimal colourings of H and satisfies k-1 < chi*(H) \le k, where k is the chromatic number of H.