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On the Decay of Crossing Numbers

  • Jacob Fox
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

Abstract

The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that \({\rm cr}(G-e) \geq k-c\sqrt{k}\). They showed only that G always has an edge e with \({\rm cr}(G-e) \geq \frac{2}{5}{\rm cr}(G)-O(1)\). We prove that for every fixed ε > 0, there is a constant n 0 depending on ε such that if G is a graph with n > n 0 vertices and m > n 1 + ε edges, then G has a subgraph G′ with at most \((1-\frac{1}{24\epsilon})m\) edges such that \({\rm cr}(G') \geq (\frac{1}{28}-o(1)){\rm cr}(G)\).

Keywords

Connected Graph Decomposition Algorithm Degree Sequence Delete Edge Crossing Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [ACNS82]
    Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. In: Theory and Practice of Combinatorics. Mathematical Studies, vol. 60, pp. 9–12. North-Holland, Amsterdam (1982)Google Scholar
  2. [ES82]
    Erdős, P., Simonovits, M.: Compactness results in extremal graph theory. Combinatorica 2, 275–288 (1982)CrossRefMathSciNetGoogle Scholar
  3. [GM90]
    Gazit, H., Miller, G.L.: Planar separators and the Euclidean norm. In: Asano, T., Imai, H., Ibaraki, T., Nishizeki, T. (eds.) SIGAL 1990. LNCS, vol. 450, pp. 338–347. Springer, Heidelberg (1990)Google Scholar
  4. [L83]
    Leighton, T.: Complexity Issues in VLSI. MIT Press, Cambridge (1983)Google Scholar
  5. [L84]
    Leighton, T.: New lower bound techniques for VLSI. Math. Systems Theory 17, 47–70 (1984)CrossRefMathSciNetMATHGoogle Scholar
  6. [LT79]
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)CrossRefMathSciNetMATHGoogle Scholar
  7. [NI92]
    Nagamochi, H., Ibaraki, T.: A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583–596 (1992)CrossRefMathSciNetMATHGoogle Scholar
  8. [PRTT04]
    Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the Crossing Lemma by finding more crossings in sparse graphs. In: 20th ACM Symposium on Computational Geometry, pp. 68–75. ACM Press, New York (2004)Google Scholar
  9. [PSS96]
    Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. Algorithmica 16, 111–117 (1996)MathSciNetMATHCrossRefGoogle Scholar
  10. [PST06]
    Pach, J., Solymosi, J., Tardos, G.: Crossing numbers of imbalanced graphs. Lecture presented at SIAM Conf. Discrete Math., Victoria, BC (2006)Google Scholar
  11. [PST00]
    Pach, J., Spencer, J., Tóth, G.: New bounds on crossing numbers. Discrete Comput. Geom. 24, 623–644 (2000)CrossRefMathSciNetMATHGoogle Scholar
  12. [PT00]
    Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9, 194–207 (2000)MathSciNetMATHGoogle Scholar
  13. [RT93]
    Richter, B., Thomassen, C.: Minimal graphs with crossing number at least k. J. Combin. Theory Ser. B 58, 217–224 (1993)CrossRefMathSciNetMATHGoogle Scholar
  14. [S00]
    Salazar, G.: On a crossing number result of Richter and Thomassen. J. Combin. Theory Ser. B 79, 98–99 (2000)CrossRefMathSciNetMATHGoogle Scholar
  15. [SSSV97]
    Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I.: Crossing numbers: bounds and applications. In: Intuitive geometry (Budapest, 1995). Bolyai Soc. Math. Stud, vol. 6, pp. 179–206. János Bolyai Math. Soc., Budapest (1997)Google Scholar
  16. [S04]
    Székely, L.A.: A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Math. 276, 331–352 (2004)CrossRefMathSciNetMATHGoogle Scholar
  17. [S04a]
    Székely, L.A.: Short proof for a theorem of Pach, Spencer, and Tóth. In: Towards a theory of geometric graphs. Contemp. Math, vol. 342, pp. 281–283. AMS, Providence (2004)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Jacob Fox
    • 1
  • Csaba D. Tóth
    • 2
  1. 1.Department of Mathematics, Princeton University, Princeton, NJ 
  2. 2.Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 

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