Rainbow connection number of comb product of graphs

Abstract

An edge-colored graph G is called a rainbow connected if any two vertices are connected by a path whose edges have distinct colors. Such a path is called a rainbow path. The smallest number of colors required in order to make G rainbow connected is called the rainbow connection number of G . For two connected graphs G and H with v ∈ V ( H ) , the comb product between G and H , denoted by G ⊳ v H , is a graph obtained by taking one copy of G and | V ( G )| copies of H and identifying the i -th copy of H at the vertex v to the i -th vertex of G . In this paper, we give sharp lower and upper bounds for the rainbow connection number of comb product between two connected graphs. We also determine the exact values of rainbow connection number of G ⊳ v H for some connected graphs G and H .