Rainbow vertex connection number of square, glue, middle and splitting graph of brush graph

Abstract

A vertex-colored graph G = (V(G), E(G)) is said a rainbow vertex-connected, if for every two vertices u and v in V(G), there exist a u−v path with all internal vertices have distinct colors. The rainbow vertex-connection number of G, denoted by rvc(G), is the smallest number of colors needed to make G rainbow vertex-connected. Let n is integers at least 2, Bn is a brush graph with 2n vertices. In this paper, we determine the rainbow vertex connection number of square, glue, middle and splitting graph of brush graph.