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IJCCGGT 2003: Combinatorial Geometry and Graph Theory pp 216-225

# Δ-Optimum Exclusive Sum Labeling of Certain Graphs with Radius One

• Mauritsius Tuga
• Mirka Miller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3330)

## Abstract

A mapping L is called a sum labeling of a graph H(V(H),E(H)) if it is an injection from V(H) to a set of positive integers, such that xyE(H) if and only if there exists a vertex wV(H) such that L(w) = L(x) + L(y). In this case, w is called a working vertex. We define L as an exclusive sum labeling of a graph G if it is a sum labeling of $$G\cup\bar K_{r}$$ for some non negative integer r, and G contains no working vertex. In general, a graph G will require some isolated vertices to be labeled exclusively. The least possible number of such isolated vertices is called exclusive sum number of G; denoted by ε(G).

An exclusive sum labeling of a graph G is said to be optimum if it labels G exclusively by using ε(G) isolated vertices. In case ε (G) = Δ (G), where Δ(G) denotes the maximum degree of vertices in G, the labeling is called Δ-optimum exclusive sum labeling.

In this paper we present Δ-optimum exclusive sum labeling of certain graphs with radius one, that is, graphs which can be obtained by joining all vertices of an integral sum graph to another vertex. This class of graphs contains infinetely many graphs including some populer graphs such as wheels, fans, friendship graphs, generalised friendship graphs and multicone graphs.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2005

## Authors and Affiliations

• Mauritsius Tuga
• 1
• Mirka Miller
• 2
1. 1.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia
2. 2.School of Information Technology and Mathematical ScienceUniversity of BallaratAustralia

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