Discussiones Mathematicae Graph Theory 27(1) (2007) 105-123
doi: 10.7151/dmgt.1348

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LINEAR AND CYCLIC RADIO k-LABELINGS OF TREES

Mustapha Kchikech,  Riadh Khennoufa and Olivier Togni

LE2I, UMR CNRS 5158
Université de Bourgogne
21078 Dijon cedex, France
e-mail: {kchikech, khennoufa, otogni}@u-bourgogne.fr

Abstract

Motivated by problems in radio channel assignments, we consider radio k-labelings of graphs. For a connected graph G and an integer k ≥ 1, a linear radio k-labeling of G is an assignment f of nonnegative integers to the vertices of G such that
|f(x)−f(y)| ≥ k+1−dG(x,y),

for any two distinct vertices x and y, where dG(x,y) is the distance between x and y in G. A cyclic k-labeling of G is defined analogously by using the cyclic metric on the labels. In both cases, we are interested in minimizing the span of the labeling. The linear (cyclic, respectively) radio k-labeling number of G is the minimum span of a linear (cyclic, respectively) radio k-labeling of G.

In this paper, linear and cyclic radio k-labeling numbers of paths, stars and trees are studied. For the path Pn of order n ≤ k+1, we completely determine the cyclic and linear radio k-labeling numbers. For 1 ≤ k ≤ n−2, a new improved lower bound for the linear radio k-labeling number is presented. Moreover, we give the exact value of the linear radio k-labeling number of stars and we present an upper bound for the linear radio k-labeling number of trees.

Keywords: graph theory, radio channel assignment, cyclic and linear radio k-labeling.

2000 Mathematics Subject Classification: 05C15, 05C78.

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Received 12 December 2005
Revised 26 August 2006