ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 28(1) (2008) 165-178
DOI: 10.7151/dmgt.1399


Mustapha Kchikech, Riadh Khennoufa and  Olivier Togni

Université de Bourgogne, 21078 Dijon cedex, France
e-mail: {kchikech, khennoufa, otogni}


Frequency planning consists in allocating frequencies to the transmitters of a cellular network so as to ensure that no pair of transmitters interfere. We study the problem of reducing interference by modeling this by a radio k-labeling problem on graphs: For a graph G and an integer k ≥ 1, a radio k-labeling of G is an assignment f of non negative integers to the vertices of G such that
|f(x)−f(y)| ≥ k+1−dG(x,y),

for any two vertices x and y, where dG(x,y) is the distance between x and y in G. The radio k-chromatic number is the minimum of max{f(x)−f(y):x,y ∈ V(G)} over all radio k-labelings f of G. In this paper we present the radio k-labeling for the Cartesian product of two graphs, providing upper bounds on the radio k-chromatic number for this product. These results help to determine upper and lower bounds for radio k-chromatic numbers of hypercubes and grids. In particular, we show that the ratio of upper and lower bounds of the radio number and the radio antipodal number of the square grid is asymptotically [3/2].

Keywords: graph theory, radio channel assignment, radio k-labeling, Cartesian product, radio number, antipodal number.

2000 Mathematics Subject Classification: 05C15, 05C78.


[1] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of cycles, Congr. Numer. 144 (2000) 129-141.
[2] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of graphs, Math. Bohem. 127 (2002) 57-69.
[3] G. Chartrand, L. Nebeský and P. Zhang, Radio k-colorings of paths, Discuss. Math. Graph Theory 24 (2004) 5-21, doi: 10.7151/dmgt.1209.
[4] G. Fertin, E. Godard and A. Raspaud, Acyclic and k-distance coloring of the grid, Inform. Process. Lett. 87 (2003) 51-58, doi: 10.1016/S0020-0190(03)00232-1.
[5] W. Imrich and S. Klavžar, Product graphs, Structure and recognition, With a foreword by Peter Winkler, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley-Interscience, New York, 2000).
[6] M. Kchikech, R. Khennoufa and O. Togni, Linear and cyclic radio k-labelings of trees, Discuss. Math. Graph Theory 27 (2007) 105-123, doi: 10.7151/dmgt.1348.
[7] R. Khennoufa and O. Togni, A note on radio antipodal colourings of paths, Math. Bohemica 130 (2005) 277-282.
[8] R. Khennoufa and O. Togni, The Radio Antipodal Number of the Hypercube, submitted, 2007.
[9] D. Král, L.-D. Tong and X. Zhu, Upper Hamiltonian numbers and Hamiltonian spectra of graphs, Australasian J. Combin. 35 (2006) 329-340.
[10] D. Liu, Radio Number for Trees, manuscript, 2006.
[11] D. Liu and M. Xie, Radio Number for Square Paths, Discrete Math., to appear.
[12] D. Liu and X. Zhu, Multi-level distance labelings for paths and cycles, SIAM J. Discrete Math. 19 (2005) 610-621, doi: 10.1137/S0895480102417768.

Received 28 March 2007
Revised 24 September 2007
Accepted 24 September 2007