Total rainbow connection number of-centipede graph and its line, square, and middle graph

Abstract

LetG= (V(G),E(G)) is a nontrivial and connected graph. A pathPatGconnecting two verticesuandvin a total-colored graphGis said to be a total-rainbow path betweenuandvif all elements inV(P) E(P), except foruandv, are assigned different colors. The total-colored graphGis said to be a rainbow connected if it has a total-rainbow total-path between every two vertices. The total-rainbow connected number of a graphGdenoted bytrc(G) is the smallest number of colors that needed to make the graphGbe a total-rainbow connected. In this paper, we determine the exact value of trc(G) whereGare Line, Square, and Middle ofn-Centipede graph.n-Centipede graph is a graph of 2nvertices obtained by joining the bottoms ofn-copies of the path graphP2laid in a row with edge and it is denoted by Cn.