Frequency Assignment and Multicoloring Powers of Square and Triangular Meshes

  • Mustapha Kchikech
  • Olivier Togni
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3503)


The static frequency assignment problem on cellular networks can be abstracted as a multicoloring problem on a weighted graph, where each vertex of the graph is a base station in the network, and the weight associated with each vertex represents the number of calls to be served at the vertex. The edges of the graph model interference constraints for frequencies assigned to neighboring stations. In this paper, we first propose an algorithm to multicolor any weighted planar graph with at most \(\frac{11}{4}W\) colors, where W denotes the weighted clique number. Next, we present a polynomial time approximation algorithm which garantees at most 2W colors for multicoloring a power square mesh. Further, we prove that the power triangular mesh is a subgraph of the power square mesh. This means that it is possible to multicolor the power triangular mesh with at most 2W colors, improving on the known upper bound of 4W. Finally, we show that any power toroidal mesh can be multicolored with strictly less than 4W colors using a distributed algorithm.


Graph multioloring power graph approximation algorithm distributed algorithm frequency assignment cellular networks 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreou, M.I., Nikoletseas, S.E., Spirakis, P.G.: Algorithms and Experiments on Colouring Squares of Planar Graphs. In: Jansen, K., Margraf, M., Mastrolli, M., Rolim, J.D.P. (eds.) WEA 2003. LNCS, vol. 2647, pp. 15–32. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Caragiannis, I., Kaklamanis, C., Papaioannou, E.: Efficient On-Line Frequency Allocation and Call Control in Cellular Networks. Proc. Theory of Computing Systems 35(5), 521–543 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Feder, T., Shende, S.M.: Online channel allocation in FDMA networks with reuse constraints. Inform. Process. Lett. 67(6), 295–302 (1998)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Janssen, J., Kilakos, K., Marcotte, O.: Fixed preference frequency allocation for cellular telephone systems. IEEE Transactions on Vehicular Technology 48(2), 533–541 (1999)CrossRefGoogle Scholar
  5. 5.
    Janssen, J., Krizanc, D., Narayanan, L., Shende, S.: Distributed On–Line Frequency Assignment in Cellular Networks. Journal of Algorithms 36(2), 119–151 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jordan, S., Schwabe, E.J.: Worst-case preference of cellular channel assignment policies. Wireless networks 2, 265–275 (1996)CrossRefGoogle Scholar
  7. 7.
    McDiarmid, C., Reed, B.: Channel Assignment and Weighted coloring. Networks 36, 114–117 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Molloy, M., Salavatipour, M.R.: Frequency Channel Assignment on Planar Networks. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 736–747. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Narayanan, L., Shende, S.M.: Static Frequency Assignment In Cellular Networks. Algoritmica 29, 396–409 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Narayanan, L., Tang, Y.: Worst-case analysis of a dynamic channel assignment strategy. Discrete Applied Mathematics 140, 115–141 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four-colour theorem. Journal of Combinatorial Theory 70(1), 2–44 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Salavatipour, M.R.: The Three color problem for planar graphs. Technical Report CSRG-458, Department of Computer Science, University of Toronto (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Mustapha Kchikech
    • 1
  • Olivier Togni
    • 1
  1. 1.LE2I, UMR CNRSUniversité de BourgogneDijon CedexFRANCE

Personalised recommendations