Discussiones Mathematicae Graph Theory 30(2) (2010) 201-222
doi: 10.7151/dmgt.1487

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Ruxandra Marinescu-Ghemeci

Faculty of Mathematics and Computer Science
University of Bucharest
Str. Academiei, 14, 010014 Bucharest, Romania
e-mail: verman@fmi.unibuc.ro


For a graph G and any two vertices u and v in G, let d(u, v) denote the distance between u and v and let diam(G) be the diameter of G. A multilevel distance labeling (or radio labeling) for G is a function f that assigns to each vertex of G a positive integer such that for any two distinct vertices u and v, d(u,v)+ | f(u)-f(v) | ≥ diam(G)+1. The largest integer in the range of f is called the span of f and is denoted span(f). The radio number of G, denoted rn(G), is the minimum span of any radio labeling for G. A thorn graph is a graph obtained from a given graph by attaching new terminal vertices to the vertices of the initial graph. In this paper the radio numbers for two classes of thorn graphs are determined: the caterpillar obtained from the path Pn by attaching a new terminal vertex to each non-terminal vertex and the thorn star Sn,k obtained from the star Sn by attaching k new terminal vertices to each terminal vertex of the star.

Keywords: multilevel distance labeling, radio number, caterpillar, diameter.

2010 Mathematics Subject Classification: 05C15, 05C78.


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Received 12 November 2008
Revised 8 June 2009
Accepted 8 June 2009