On the Rainbow Vertex Connection Number of Edge Comb of Some Graph

Abstract

By an edge comb, we mean a graph formed by combining two graphs G and H, where each edge of graph G is replaced by the which one edge of graph H, denote by G D H. A vertex colored graph G D H = (V (G D H), E(G D H)) is said rainbow vertex-connected, if for every two vertices u and v in V (G D H), there is a u − v path with all internal vertices have distinct color. The rainbow vertex connection number of G D H, denoted by rvc(G D H) is the smallest number of color needed in order to make G D H rainbow vertex-connected. This research aims to find an exact value of the rainbow vertex connection number of exponential graph, namely rvc(G D H) when G D H are Pn D Btm, Sn D Btm, Ln D Btm, Fm,n D Btp, rvc(Pn D Sm), rvc(Cn D Sm), and rvc(Wn D Sm) Wn D Btm. The result shows that the resulting rainbow vertex connection attain the given lower bound