Rainbow Numbers for Cycles with Pendant Edges

Abstract

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f (n, H) is the maximum number of colors in an edge-coloring of K n with no rainbow copy of H. The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of K n with rb(n, H) number of colors contains a rainbow copy of H. Certainly rb(n, H) = f(n, H) + 1. Anti-Ramsey numbers were introduced by Erdős et al. [4] and studied in numerous papers. We show that $$rb(n, C_k^+) = rb(n, C_k)$$for n ≥ k + 1, where C k + denotes a cycle C k with a pendant edge.