Ordered Coloring Grids and Related Graphs

  • Amotz Bar-Noy
  • Panagiotis Cheilaris
  • Michael Lampis
  • Valia Mitsou
  • Stathis Zachos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5869)

Abstract

We investigate a coloring problem, called ordered coloring, in grids and some other families of grid-like graphs. Ordered coloring (also known as vertex ranking) is related to conflict-free coloring and other traditional coloring problems. Such coloring problems can model (among others) efficient frequency assignments in cellular networks. Our main technical results improve upper and lower bounds for the ordered chromatic number of grids and related graphs. To the best of our knowledge, this is the first attempt to calculate exactly the ordered chromatic number of these graph families.

Keywords

grid graph ordered coloring vertex ranking conflict-free coloring 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bar-Noy, A., Cheilaris, P., Smorodinsky, S.: Deterministic conflict-free coloring for intervals: from offline to online. ACM Transactions on Algorithms 4(4), 44:1–44:18 (2008)Google Scholar
  2. 2.
    Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms 18(2), 238–255 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chen, K., Fiat, A., Kaplan, H., Levy, M., Matoušek, J., Mossel, E., Pach, J., Sharir, M., Smorodinsky, S., Wagner, U., Welzl, E.: Online conflict-free coloring for intervals. SIAM Journal on Computing 36(5), 1342–1359 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Deogun, J.S., Kloks, T., Kratsch, D., Müller, H.: On the vertex ranking problem for trapezoid, circular-arc and other graphs. Discrete Applied Mathematics 98, 39–63 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Elbassioni, K., Mustafa, N.H.: Conflict-free colorings of rectangles ranges. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 254–263. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Even, G., Lotker, Z., Ron, D., Smorodinsky, S.: Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM Journal on Computing 33, 94–136 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Har-Peled, S., Smorodinsky, S.: Conflict-free coloring of points and simple regions in the plane. Discrete and Computational Geometry 34, 47–70 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Iyer, A.V., Ratliff, H.R., Vijayan, G.: Optimal node ranking of trees. Information Processing Letters 28, 225–229 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Katchalski, M., McCuaig, W., Seager, S.: Ordered colourings. Discrete Mathematics 142, 141–154 (1995)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Leiserson, C.E.: Area-efficient graph layouts (for VLSI). In: Proceedings of the 21st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 270–281 (1980)Google Scholar
  11. 11.
    Liu, J.W.H.: The role of elimination trees in sparse factorization. SIAM Journal on Matrix Analysis and Applications 11(1), 134–172 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Llewellyn, D.C., Tovey, C.A., Trick, M.A.: Local optimization on graphs. Discrete Applied Mathematics 23(2), 157–178 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pach, J., Tóth, G.: Conflict free colorings. In: Discrete and Computational Geometry, The Goodman-Pollack Festschrift, pp. 665–671. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Smorodinsky, S.: Combinatorial Problems in Computational Geometry. PhD thesis, School of Computer Science, Tel-Aviv University (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Amotz Bar-Noy
    • 1
    • 2
  • Panagiotis Cheilaris
    • 3
  • Michael Lampis
    • 2
  • Valia Mitsou
    • 2
  • Stathis Zachos
    • 1
    • 2
    • 4
  1. 1.Computer and Information Science Department Brooklyn CollegeCity University of New YorkBrooklynUSA
  2. 2.Doctoral Program in Computer Science The Graduate CenterCity University of New YorkUSA
  3. 3.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary
  4. 4.School of Electrical and Computer EngineeringNational Technical University of AthensAthensGreece

Personalised recommendations