Bounds and Methods for k-Planar Crossing Numbers

  • Farhad Shahrokhi
  • Ondrej Sýkora
  • Laszlo A. Székely
  • Imrich Vrt’o
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)


The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K2 k + 1, q, for k ≥ 2. We prove tight bounds for complete graphs.


  1. 1.
    Aggarwal, A., Klawe, M., Shor, P.: Multi-layer grid embeddings for VLSI. Algorithmica 6, 129–151 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beineke, L.W.: Complete bipartite graphs: Decomposition into planar subgraphs. In: Harary, F. (ed.) A Seminar on Graph Theory, Selected Topics in Mathematics. ch. 7, pp. 43–53. Holt, Rinehart and Winston (1967)Google Scholar
  3. 3.
    Beineke, L.W.: Biplanar graphs: A survey. Computers and Mathematics with Applications 34, 1–8 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Beineke, L.W., Harary, F., Moon, J.W.: On the thickness of the complete bipartite graphs. In: Proc. of the Cambridge Philosophical Society, vol. 60, pp. 1–5 (1964)Google Scholar
  5. 5.
    Czabarka, É., Sýkora, O., Székely, L.A., Vrťo, I.: Biplanar crossing numbers: A survey of results and problems. In: Fleiner, T., Katona, G.O.H. (eds.) Finite and Infinite Combinatorics, Akadémia Kiadó, Budapest. Bolyai Society Mathematical Studies (to appear)Google Scholar
  6. 6.
    Kleitman, D.J.: The crossing number of K.,n. J. Combinatorial Theory 9, 315–323 (1970)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Leighton, T.F.: Complexity Issues in VLSI. MIT Press, Cambridge (1983)Google Scholar
  8. 8.
    Nash-Williams, J.A.: Edge disjoint spanning trees of finite graphs. J. London Math. Soc. 36, 445–450 (1961)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Owens, A.: On the biplanar crossing number. IEEE Transactions on Circuit Theory 18, 277–280 (1971)CrossRefGoogle Scholar
  10. 10.
    Richter, R.B., Širáň, J.: The crossing number of K.,n in a surface. J. Graph Theory 21, 51–54 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrťo, I.: The book crossing number of graphs. J. Graph Theory 21, 413–424 (1996)Google Scholar
  12. 12.
    Sýkora, O., Székely, L.A., Vrťo, I.: Crossing numbers and biplanar crossing numbers: using the probabilistic method (submitted)Google Scholar
  13. 13.
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrťo, I.: Bounds for convex crossing numbers. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 487–495. Springer, Heidelberg (2003)Google Scholar
  14. 14.
    Truszczyński, M.: Decomposition of graphs into forests with bounded maximum degree. Discrete Mathematics 98, 207–222 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    White, A.T., Beineke, L.W.: Topological graph theory. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory, pp. 15–50. Academic Press, New York (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Farhad Shahrokhi
    • 1
  • Ondrej Sýkora
    • 2
  • Laszlo A. Székely
    • 3
  • Imrich Vrt’o
    • 4
  1. 1.Department of Computer ScienceUniversity of North TexasDentonUSA
  2. 2.Department of Computer ScienceLoughborough University LoughboroughLeicestershireThe United Kingdom
  3. 3.Department of MathematicsUniversity of South CarolinaColumbiaUSA
  4. 4.Department of InformaticsInstitute of Mathematics Slovak Academy of SciencesBratislavaSlovak Republic

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