On total irregularity strength of caterpillar graphs with two leaves on each internal vertex
Isnaini Anniswati Rosyida, Nashi Widodo, Diari Indriati · Journal of Physics Conference Series · 2018
Let G ( V, E ) be a graph. A function f from to the set {1, 2, ..., k } is said to be a totally irregular total k -labeling of G if the weights of any two different vertices x and y in V ( G ) satisfy where the weight w f ( x ) is the sum of label of x and labels of all edges incident to x , and the weights of any two different edges ux and vy in E ( G ) satisfy where the weight w f ( ux ) is obtained from the sum of: label of x , label of u and label of edge ux . The total irregularity strength of the graph G , denoted by ts ( G ), is the minimum number k for which G has a totally irregular total k -labeling. In this paper, we focus on a caterpillar graph with two leaves on each internal vertex T 2 n + p,q , where n is the number of leaves on each end vertex of the central path, p is the number of leaves connected to internal vertices, q is the number of vertices of the central path, p > 4 and q > 4. Firstly, we do some experiments for constructing a formula for totally irregular total k -labeling of the caterpillar graph T 2 n + p,q . Secondly, we determine the minimum number k which is ts of the caterpillar graph T 2 n + p,q . We obtain that and .