Asymmetric Ramsey properties of random graphs involving cliques

Consider the following problem: For given graphs G and F1,,Fk, find a coloring of the edges of G with k colors such that G does not contain Fi in color i. Rödl and Ruciski studied this problem for the random graph Gn,p in the symmetric case when k is fixed and F1 = ··· = Fk = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p bn- for some constants b = b(F,k) and = (F). This result is essentially best possible because for p Bn-, where B = B(F,k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n-(F1,,Fk) for arbitrary F1,,Fk. In this article we address the case when F1,,Fk are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of Gn,p with p bn- for some constant b = b(F1,,Fk), where = (F1,,Fk) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F1,,Fk) such that for p Bn- the random graph Gn,p a.a.s. does not have a valid k-edge-coloring provided the so-called KR-conjecture holds. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009