On total irregularity strength of caterpillar graphs with two leaves on each internal vertex

Abstract

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 wf(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 wf(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 T2n+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 T2n+p,q. Secondly, we determine the minimum number k which is ts of the caterpillar graph T2n+p,q. We obtain that and .