Anti-Ramsey problems for cycles

Abstract

We call a subgraph of an edge-colored graph rainbow, if all of its edges have different colors. A rainbow copy of a graph H in an edge-colored graph G is a subgraph of G isomorphic to H such that the coloring restricted to this subgraph is a rainbow coloring. Given two graphs G and H, let Ar(G,H) denote the maximum number of colors in a coloring of the edges of G that has no any rainbow copy of H. When G is Kn, Ar(G,H) is called the anti-Ramsey number. Anti-Ramsey number was introduced by Erdös, Simonovits and Sós in the 1970s. Since then, the field has blossomed in a wide variety of papers and some other graphs were used as host graphs. In this paper, we give a new method to compute the anti-Ramsey number by constructing a corresponding hypergraph. As application, we determine the exact values of Ar(G,Ck) when the host graph G is Wd, Pm×Pn, Pm×Cn, Cm×Cn and cyclic Cayley graph, respectively.