Total Rainbow Connected Numbers for Ladder and Steering Amalgamation Graphs J Mucklow

Abstract

All graph considered in this paper are finite, simple, and undirected. Let  be a nontrivial connected graph and  be a natural number. A mapping  is called a rainbow total--coloring if any two vetherertices have distinct colors. The path like that is called a rainbow total-path. Graph  is called rainbow total-connected if any two vertices  and  in  there exist a rainbow total-path . The rainbow total-connection number of , denoted by , is the smallest number of colors needed to make  rainbow total-connected. Let  be a natural number with 2. Let  be a finite collection of graph and each   has fixed vertex  called a terminal. The amalgamation  is a graph formed by taking all the  ’s and identifying their terminals. In this paper is determined lower and upper bounds for the total-rainbow connection number of an amalgamation graph. Additionally, we determine the total-rainbow connection number of amalgamation of ladders and helms.