Skip to main content

A Crossing Lemma for the Pair-Crossing Number

  • Conference paper
  • 1455 Accesses

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8871)

Abstract

The pair-crossing number of a graph G, pcr(G), is the minimum possible number of pairs of edges that cross each other (possibly several times) in a drawing of G. It is known that there is a constant c ≥ 1/64 such that for every (not too sparse) graph G with n vertices and m edges \({\mbox{pcr}}(G) \geq c \frac{m^3}{n^2}\). This bound is tight, up to the constant c. Here we show that c ≥ 1/34.2 if G is drawn without adjacent crossings.

References

  1. Ackerman, E.: On topological graphs with at most four crossings per edge (manuscript)

    Google Scholar 

  2. Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Combinatorial Theory, Ser. A 114(3), 563–571 (2007)

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Aigner, M., Ziegler, G.: Proofs from the Book. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  4. Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. In: Theory and Practice of Combinatorics. North-Holland Math. Stud., vol. 60, pp. 9–12. North-Holland, Amsterdam (1982)

    Google Scholar 

  5. Cranston, D.W., West, D.B.: A guide to discharging (manuscript)

    Google Scholar 

  6. Leighton, F.T.: Complexity Issues in VLSI: Optimal Layouts for the Shuffle-Exchange Graph and Other Networks. MIT Press, Cambridge (1983)

    Google Scholar 

  7. Matoušsek, J.: Near-optimal separators in string graphs, ArXiv (February 2013)

    Google Scholar 

  8. Montaron, B.: An improvement of the crossing number bound. J. Graph Theory 50(1), 43–54 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the crossing lemma by finding more crossings in sparse graphs. Disc. Compu. Geometry 36(4), 527–552 (2006)

    CrossRef  MATH  Google Scholar 

  10. Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9(4), 194–207 (2000)

    MATH  MathSciNet  Google Scholar 

  12. Pach, J., Tóth, G.: Which crossing number is it anyway? J. Combin. Theory Ser. B 80(2), 225–246 (2000)

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing independently even crossings. SIAM J. on Disc. Math. 24(2), 379–393 (2010)

    CrossRef  MATH  Google Scholar 

  14. Schaefer, M.: The graph crossing number and its variants: A survey. Electronic Journal of Combinatorics, Dynamic Survey 21 (2013)

    Google Scholar 

  15. Schaefer, M., Štefankovič, D.: Decidability of string graphs. J. Comput. System Sci. 68(2), 319–334 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Ackerman, E., Schaefer, M. (2014). A Crossing Lemma for the Pair-Crossing Number. In: Duncan, C., Symvonis, A. (eds) Graph Drawing. GD 2014. Lecture Notes in Computer Science, vol 8871. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45803-7_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-45803-7_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-45802-0

  • Online ISBN: 978-3-662-45803-7

  • eBook Packages: Computer ScienceComputer Science (R0)