Discussiones Mathematicae Graph Theory 30(4) (2010) 539-544
doi: 10.7151/dmgt.1511

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ARITHMETIC LABELINGS AND GEOMETRIC LABELINGS OF COUNTABLE GRAPHS

Gurusamy Rengasamy Vijayakumar

School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba, Mumbai 400 005, India
e-mail: vijay@math.tifr.res.in

Abstract

An injective map from the vertex set of a graph G-its order may not be finite-to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is called arithmetic (geometric) if it admits an arithmetic (a geometric) labeling. In this article, we show that the two notions just mentioned are equivalent-i.e., a graph is arithmetic if and only if it is geometric.

Keywords: arithmetic labeling of a graph, geometric labeling of a graph.

2010 Mathematics Subject Classification: 05C78, 05C63.

References

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Received 14 February 2009
Revised 19 October 2009
Accepted 20 October 2009