Radio number for some thorn graphs

Ruxandra Marinescu-Ghemeci

Discussiones Mathematicae Graph Theory (2010)

  • Volume: 30, Issue: 2, page 201-222
  • ISSN: 2083-5892

Abstract

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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 S n , k obtained from the star Sₙ by attaching k new terminal vertices to each terminal vertex of the star.

How to cite

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Ruxandra Marinescu-Ghemeci. "Radio number for some thorn graphs." Discussiones Mathematicae Graph Theory 30.2 (2010): 201-222. <http://eudml.org/doc/271026>.

@article{RuxandraMarinescu2010,
abstract = {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 $S_\{n,k\}$ obtained from the star Sₙ by attaching k new terminal vertices to each terminal vertex of the star.},
author = {Ruxandra Marinescu-Ghemeci},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {multilevel distance labeling; radio number; caterpillar; diameter},
language = {eng},
number = {2},
pages = {201-222},
title = {Radio number for some thorn graphs},
url = {http://eudml.org/doc/271026},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Ruxandra Marinescu-Ghemeci
TI - Radio number for some thorn graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 2
SP - 201
EP - 222
AB - 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 $S_{n,k}$ obtained from the star Sₙ by attaching k new terminal vertices to each terminal vertex of the star.
LA - eng
KW - multilevel distance labeling; radio number; caterpillar; diameter
UR - http://eudml.org/doc/271026
ER -

References

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  1. [1] F. Buckley and F. Harary, Distance in Graphs (Addison-Wesley Pub. Co., 1990). Zbl0688.05017
  2. [2] G. Chartrand, D. Erwin, F. Harary and P. Zhang, Radio labelings of graphs, Bull. Inst. Combin. Appl. 33 (2001) 77-85. Zbl0989.05102
  3. [3] G. Chartrand, D. Erwin and P. Zhang, A graph labeling problem suggested by FM channel restrictions, Bull. Inst. Combin. Appl. 43 (2005) 43-57. Zbl1066.05125
  4. [4] J.A. Gallian, A dynamic survey of graph labelings, Electronic J. Combinatorics 14 (2007) #DS6. 
  5. [5] I. Gutman, Distance of thorny graphs, Publ. Inst. Math. (Beograd) 63 (1998) 31-36. Zbl0942.05021
  6. [6] W.K. Hale, Frequency assignment: theory and applications, Proc. IEEE 68 (1980) 1497-1514, doi: 10.1109/PROC.1980.11899. 
  7. [7] D. Liu, Radio number for trees, Discrete Math. 308 (2008) 1153-1164, doi: 10.1016/j.disc.2007.03.066. Zbl1133.05090
  8. [8] D. Liu and X. Zhu, Multilevel distance labelings for paths and cycles, SIAM J. Discrete Math. 19 (2005) 610-621, doi: 10.1137/S0895480102417768. Zbl1095.05033
  9. [9] M.T. Rahim and I. Tomescu, Multilevel distance labelings for helm graphs, Ars Combinatoria, to appear. 

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