On the Total Edge Irregularity Strength of Disjoint Union of Graphs

Abstract

For a simple graph G=(V(G),E(G)), a total labeling f: V(G)jE(G) g{1,2, ... ,k} is called k-labeling. The weight of an edge xy in G, denoted by wtf(xy), is the sum of the edge label itself and the labels of end vertices x and y, i.e. wtf(xy)=f(x)+f(xy)+f(y). A total k-labeling is defined to be an edge irregular total k-labeling of the graph G if for every two different edges xy and x′y′ there is wtf(xy)≠wtf(x′y′). The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). In this paper, we estimate the upper bound of the total edge irregularity strength of disjoint union of multiple copies of a graph and we prove that this upper bound is tight.