THE RAINBOW TOTAL-CONNECTION NUMBER OF AMALGAMATION OF SOME GRAPHS - Arbain
All graph considered in this thesis are finite, simple, and undirected. Let G = (V (G);E(G)) be a nontrivial connected graph and k be a natural number. A mapping c : V (G) U E(G) ->f(1; 2; .... ; k) is called a rainbow total-k-coloring if any two vertices x and y in V (G) there exists a x-y path with all edges and internal vertices have distinct colors. The path like that is called a rainbow total-path. Graph G is called rainbow total-connected if any two vertices x and y in V (G) there exist a rainbow total-path x - y. The rainbow total- connection number of G, denoted by trc(G), is the smallest number of colors needed to make G rainbow total-connected. Let t be a natural number with t - 2. Let fGiji 2 [1; t]g be a finite collection of graphs and each Gi has a fixed vertex v0i called a terminal. The amalgamation Amal(Gi; v0i; t) is a graph formed by taking all the Gi 's and identifying their terminals. In this thesis 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 trees, ladders, helms, and complete graphs.