Radio k-labelings for Cartesian products of graphs

• Autorzy: Mustapha Kchikech , Riadh Khennoufa , Olivier Togni
• Języki publikacji: EN

Abstrakty

EN

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 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].

Słowa kluczowe

EN

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

Kategorie tematyczne

• 05C15: Coloring of graphs and hypergraphs
• 05C78: Graph labelling (graceful graphs, bandwidth, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2008
• Tom: 28
• Numer: 1
• Strony: 165-178

Daty

wydano

2008

otrzymano

2007-03-28

poprawiono

2007-09-24

zaakceptowano

2007-09-24

Twórcy

autor

Mustapha Kchikech

• LE2I, UMR CNRS 5158, Université de Bourgogne, 21078 Dijon cedex, France

autor

Riadh Khennoufa

• LE2I, UMR CNRS 5158, Université de Bourgogne, 21078 Dijon cedex, France

autor

Olivier Togni

• LE2I, UMR CNRS 5158, Université de Bourgogne, 21078 Dijon cedex, France

Bibliografia

• [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.

Identyfikatory

DOI

10.7151/dmgt.1399