Edge-colorings avoiding rainbow and monochromatic subgraphs

For two graphs G and H, let the mixed anti-Ramsey numbers, max R(n: G, H), (min R(n: G, H)) be the maximum (minimum) number of colors used in an edge-coloring of a complete graph with it vertices having no monochromatic subgraph isomorphic to G and no totally multicolored (rainbow) subgraph isomorphic to H. These two numbers generalize the classical anti-Ramsey and Ramsey numbers, respectively. We show that max R(n: G. H), in most cases, can be expressed in terms of vertex arboricity of H and it does not depend on the graph G. In particular, we determine max R(n; G, H) asymptotically for all graphs G and H, where G is not a star and H has vertex arboricity at least 3. In studying min R(n: G, H) we primarily concentrate on the case when G = H = K-3. We find min R(n K-3, K-3) exactly, as well as all extremal colorings. Among others. by investigating min R(n: K-t. K-3), we show that if an edge-coloring of K-n in k colors has no monochromatic K-t and no rainbow triangle, then n <= 2(kt2). (C) 2007 Elsevier B.V. All rights reserved.