ON COLORINGS AVOIDING A RAINBOW CYCLE AND A FIXED MONOCHROMATIC SUBGRAPH

Let H and G be two graphs on fixed number of vertices. An edge coloring of a complete graph is called (H,G)-good if there is no monochromatic copy of G and no rainbow (totally multicolored) copy of H in this coloring. As shown by Jamison and West, an (H, G)-good coloring of an arbitrarily large complete graph exists unless either G is a star or H is a forest. The largest number of colors in an (H, G)-good coloring of K-n is denoted maxR(n, G, H). For graphs H which can not be vertex-partitioned into at most two induced forests, maxR(n,G, H) has been determined asymptotically. Determining maxR(n;G,H) is challenging for other graphs H, in particular for bipartite graphs or even for cycles. This manuscript treats the case when H is a cycle. The value of maxR(n,G,C-k) is determine for all graphs G whose edges do not induce a star.