Edge-colorings avoiding rainbow and monochromatic subgraphs

Abstract

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 n 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 kt 2 .