Discussiones Mathematicae Graph Theory 27(2) (2007) 251-268
doi: 10.7151/dmgt.1359

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FURTHER RESULTS ON SEQUENTIALLY ADDITIVE GRAPHS

Suresh Manjanath Hegde

Department of mathematical and Computational sciences
National Institute of Technology Karnataka
Surathkal, Srinivasnagar-575025, India
e-mail: smhegde@nitk.ac.in

Mirka Miller

School of Electrical Engineering and Computer Science
University of Newcastle
Callaghan NSW 2308, Australia
e-mail: mirka@cs.newcastle.edu.au

Abstract

Given a graph G with p vertices, q edges and a positive integer k, a k-sequentially additive labeling of G is an assignment of distinct numbers k, k+1, k+2,☐,k+p+q−1 to the p+q elements of G so that every edge uv of G receives the sum of the numbers assigned to the vertices u and v. A graph which admits such an assignment to its elements is called a k-sequentially additive graph.

In this paper, we give an upper bound for k with respect to which the given graph may possibly be k-sequentially additive using the independence number of the graph. Also, we prove a variety of results on k-sequentially additive graphs, including the number of isolated vertices to be added to a complete graph with four or more vertices to be simply sequentially additive and a construction of an infinite family of k-sequentially additive graphs from a given k-sequentially additive graph.

Keywords: simply (k-)sequentially additive labelings (graphs), segregated labelings.

2000 Mathematics Subject Classification: 05C78.

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Received 4 January 2006
Revised 5 February 2007
Accepted 5 February 2007