Advertisement

Improvement on the Decay of Crossing Numbers

  • Jakub Černý
  • Jan Kynčl
  • Géza Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)

Abstract

We prove that the crossing number of a graph decays in a “continuous fashion” in the following sense. For any ε> 0 there is a δ> 0 such that for n sufficiently large, every graph G with n vertices and m ≥ n 1 + ε edges has a subgraph G′ of at most (1 − δ)m edges and crossing number at least Open image in new window . This generalizes the result of J. Fox and Cs. Tóth.

References

  1. [AHL02]
    Alon, N., Hoory, S., Linial, N.: The Moore bound for irregular graphs. Graphs and Combinatorics 18, 53–57 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. [FT06]
    Fox, J., Tóth, Cs.: On the decay of crossing numbers. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 174–183. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. [L83]
    Leighton, T.: Complexity issues in VLSI. MIT Press, Cambridge (1983)Google Scholar
  4. [PRTT06]
    Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the Crossing Lemma by finding more crossings in sparse graphs. In: Proc. 19th ACM Symposium on Computational Geometry, pp. 68–75 (2004), Also in: Discrete and Computational Geometry 36, 527–552 (2006)Google Scholar
  5. [PT00]
    Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9, 194–207 (2000)MATHMathSciNetGoogle Scholar
  6. [RT93]
    Richter, B., Thomassen, C.: Minimal graphs with crossing number at least k. J. Combin. Theory Ser. B 58, 217–224 (1993)MATHCrossRefMathSciNetGoogle Scholar
  7. [SSSV97]
    Shahrokhi, F., Sýkora, O., Székely, L., Vrt’o, I.: Crossing numbers: bounds and applications. In: Intuitive geometry, Budapest (1995) Bolyai Soc. Math. Stud. 6, 179–206 (1997)Google Scholar
  8. [S04]
    Székely, L.: Short proof for a theorem of Pach, Spencer, and Tóth. In: Towards a theory of geometric graphs, Contemporary Mathematics, vol. 342, pp. 281–283. AMS, Providence, RI (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Jakub Černý
    • 1
  • Jan Kynčl
    • 1
  • Géza Tóth
    • 2
  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityCzech Republic
  2. 2.Hungarian Academy of SciencesRényi InstituteBudapestHungary

Personalised recommendations