Edge-colorings avoiding rainbow and monochromatic subgraphs

For two graphs G and H, let the mixed anti-Ramsey numbers, maxR(n; G, H), (minR(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 maxR(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 maxR(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 minR(n; G, H) we primarily concentrate on the case when G = H = K3. We find minR(n; K3, K3) exactly, as well as all extremal colorings. Among others, by investigating minR(n; Kt, K3), we show that if an edge-coloring of Kn in k colors has no monochromatic Kt and no rainbow triangle, then n 2kt2 .

• Type: Journal
• Year: 2008
• Where: DM
• Authors: Maria Axenovich, Perry Iverson
• Keywords: Complete Graph, DM 2008, Mixed Anti-Ramsey Numbers, Monochromatic Subgraph