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

Let G = (V(G), E(G)) is a nontrivial and connected graph. A path P at G connecting two vertices u and v in a total-colored graph G is said to be a total-rainbow path between u and v if all elements in V(P) ∪ E(P), except for u and v, are assigned different colors. The total-colored graph G is 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 graph G denoted by trc(G) is the smallest number of colors that needed to make the graph G be a total-rainbow connected. In this paper, we determine the exact value of trc(G) where G are Line, Square, and Middle of n-Centipede graph. n-Centipede graph is a graph of 2n vertices obtained by joining the bottoms of n-copies of the path graph P2 laid in a row with edge and it is denoted by Cn.