Exclusive Sum Labeling of Graphs: A Survey

Abstract

All sum graphs are disconnected. In order for a connected graph to bear a sum labeling, the graph is considered in conjunction with a number of isolated vertices, the labels of which complete the sum labeling for the disjoint union. The smallest number of isolated vertices that must be added to a graph H to achieve a sum graph is called the sum number of H; it is denoted by σ(H). A sum labeling which realizes as a sum graph is called an optimal sum labeling of H.In this paper we survey a new type of labeling based on summation, the exclusive sum, labeling. A sum labeling L is called exclusive sum labeling with respect to a subgraph H of G if L is a sum labeling of G where H contains no working vertex. The exclusive sum number ∊(H) of a graph H is the smallest number r such that there exists an exclusive sum labeling L which realizes as a sum graph. A labeling L is an optimal exclusive sum labeling of a graph H if L is a sum labeling of H ≼ K∊(H) and H contains no working vertex.