Total Edge Irregularity Strength of Cycle Snake Graphs

Abstract

Let G be a simple undirected graph with vertex and edge sets. A total labeling that assigns integers from 1 to k to the union of the vertex and edge sets is called a k-total labeling. The weight of an edge uv, denoted by w(uv), is defined as the sum of the labels of the two vertices u, v, and the label of edge uv itself. A k-total labeling is called an edge irregular total k-labeling of G if the weights of all distinct edges are different. The minimum k for which every edge of G has a distinct weight is called the total edge irregularity strength of G, denoted by tes(G). A cycle snake graph CS_m,n is obtained from the path graph P_n with n + 1 vertices and n edges by replacing each edge with a cycle graph C_m, where m ≥ 3 and n ≥ 2. In this paper, we study the graphs CS_3,n and CS_m,n and determine their total edge irregularity strength. The case m = 3 is considered separately because the structure of CS_3,n gives a different representation of the vertex and edge sets than for m ≥ 4, requiring a different labeling construction.