The total rainbow connection on comb product of cycle and path graphs

Abstract

All graphs are finite, connected, simple, non-trivial, and undirected. The graph in this study with V(G) as vertex set and E(G) as edge set. Total-colored graph G is a total connected rainbow of any two vertices which are connected by, at least, one total rainbow path which is the edges and internal vertices which have distinct colors. The total rainbow coloring is a natural extension of edge rainbow coloring and vertex rainbow coloring. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. This paper shows the lower and upper bounds of trc(G) and the result of the total rainbow connection number of comb product of cycle and path graphs.