On total edge irregularity strength of some cactus chain graphs with pendant vertices

Abstract

Let G(V, E) be a graph. A function f from \(V(G)\cup E(G)\) to the set {1, 2, …, k} is called an edge irregular total k-labeling of G if the weights of any two different edges ux and vy in E(G) satisfy \({w}_{f}(ux)\ne {w}_{f}(vy)\) where the weight w_f (ux) is equal to the sum of label of x, label of u and label of edge ux. The total edge irregularity strength of the graph G, denoted by tes(G), is the minimum number k for which G has an edge irregular total k-labeling. In this paper, we investigate the exact value of tes of triangular cactus chain graph and para square cactus chain graph. We get the total edge irregularity strength of triangular cactus chain with length r and r + 1 pendant vertices as follows: \({\rm{tes}}(T{C}_{r}^{r+1})=\lceil \frac{4r+3}{3}\rceil \) . Further, the total edge irregularity strength of para square cactus chain graph with length r and r pendant vertices is \({\rm{tes}}({Q}_{r}^{r})=\lceil \frac{5r+2}{3}\rceil \) .