On the (Strong) Rainbow Vertex Connection of Graphs Resulting from Edge Comb Product

Abstract

The vertex-colored graph G = (V, E) is said rainbow vertex-connected, if for every two vertices u and v in V, 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 required in order to make graph G to be rainbow vertex-connected. If every two vertices u and v in V are connected by at least one shortest rainbow path, then G is strongly rainbow vertex-connected. The strong rainbow vertex- connection number, denoted by srvc(G), is the minimum number of colors required in order make graph G to be strongly rainbow vertex-connected. In this paper, we determine the rainbow vertex connection number and the strong rainbow vertex connection number of graphs which are resulted from edge comb product.