Discussiones Mathematicae Graph Theory 31(1) (2011) 45-62
doi: 10.7151/dmgt.1529

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RADIO NUMBERS FOR GENERALIZED PRISM GRAPHS

Paul Martinez

California State University Channel Islands
e-mail: paul.martinez@csuci.edu

Juan Ortiz

Lehigh University
e-mail: jpo208@lehigh.edu

Maggy Tomova

The University of Iowa
e-mail: mtomova@math.uiowa.edu

Cindy Wyels

California State University Channel Islands
e-mail: cynthia.wyels@csuci.edu

Abstract

A radio labeling is an assignment c:V(G)→ N such that every distinct pair of vertices u,v satisfies the inequality d(u,v)+|c(u)−c(v)| ≥ diam(G)+1. The span of a radio labeling is the maximum value. The radio number of G, rn(G), is the minimum span over all radio labelings of G. Generalized prism graphs, denoted Zn,s, s ≥ 1, n ≥ s, have vertex set {(i,j) | i = 1,2 and j = 1,...,n} and edge set {((i,j),(i,j ±1))} ∪{((1,i),(2,i+σ)) | σ = −⌊(s−1)/2⌋ …,0,…,⌊s/2⌋}. In this paper we determine the radio number of Zn,s for s = 1,2 and 3. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.

Keywords: radio number, radio labeling, prism graphs.

2010 Mathematics Subject Classification: 05C78, 05C15.

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Received 1 April 2009
Revised 1 April 2010
Accepted 6 April 2010