# Radio k-labelings for Cartesian products of graphs

Mustapha Kchikech; Riadh Khennoufa; Olivier Togni

Discussiones Mathematicae Graph Theory (2008)

- Volume: 28, Issue: 1, page 165-178
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topMustapha Kchikech, Riadh Khennoufa, and Olivier Togni. "Radio k-labelings for Cartesian products of graphs." Discussiones Mathematicae Graph Theory 28.1 (2008): 165-178. <http://eudml.org/doc/270576>.

@article{MustaphaKchikech2008,

abstract = {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
$|f(x)-f(y)| ≥ k+1-d_G(x,y)$,
for any two vertices x and y, where $d_G(x,y)$ is the distance between x and y in G. The radio k-chromatic number is the minimum of maxf(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].},

author = {Mustapha Kchikech, Riadh Khennoufa, Olivier Togni},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {graph theory; radio channel assignment; radio k-labeling; Cartesian product; radio number; antipodal number; radio -labeling},

language = {eng},

number = {1},

pages = {165-178},

title = {Radio k-labelings for Cartesian products of graphs},

url = {http://eudml.org/doc/270576},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Mustapha Kchikech

AU - Riadh Khennoufa

AU - Olivier Togni

TI - Radio k-labelings for Cartesian products of graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2008

VL - 28

IS - 1

SP - 165

EP - 178

AB - 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
$|f(x)-f(y)| ≥ k+1-d_G(x,y)$,
for any two vertices x and y, where $d_G(x,y)$ is the distance between x and y in G. The radio k-chromatic number is the minimum of maxf(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].

LA - eng

KW - graph theory; radio channel assignment; radio k-labeling; Cartesian product; radio number; antipodal number; radio -labeling

UR - http://eudml.org/doc/270576

ER -

## References

top- [1] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of cycles, Congr. Numer. 144 (2000) 129-141. Zbl0976.05028
- [2] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of graphs, Math. Bohem. 127 (2002) 57-69. Zbl0995.05056
- [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. Zbl1056.05053
- [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. Zbl1175.68293
- [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). Zbl0963.05002
- [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. Zbl1137.05063
- [7] R. Khennoufa and O. Togni, A note on radio antipodal colourings of paths, Math. Bohemica 130 (2005) 277-282. Zbl1110.05033
- [8] R. Khennoufa and O. Togni, The Radio Antipodal Number of the Hypercube, submitted, 2007. Zbl1265.05536
- [9] D. Král, L.-D. Tong and X. Zhu, Upper Hamiltonian numbers and Hamiltonian spectra of graphs, Australasian J. Combin. 35 (2006) 329-340. Zbl1095.05023
- [10] D. Liu, Radio Number for Trees, manuscript, 2006.
- [11] D. Liu and M. Xie, Radio Number for Square Paths, Discrete Math., to appear. Zbl1224.05451
- [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. Zbl1095.05033

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.