The Rainbow (Vertex) Connection Number of Pencil Graphs

Abstract

An edge colored graph G = (V(G), E(G)) is said rainbow connected, if any two vertices are connnected by a path whose edges have distinct colors. The rainbow connection number of G, denoted by rc(G), is the smallest positive integer of colors needed in order to make G rainbow connected. The vertex-colored graph G is said rainbow vertex-connected, if for every two vertices u and v in V(G), there is a u-v path with all internal vertices have distinct color. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors needed in order to make G rainbow vertex-connected. In this paper, we determine rainbow (vertex) connection number of pencil graphs.