Discrete & Computational Geometry

, Volume 39, Issue 4, pp 791–799

Note on the Pair-crossing Number and the Odd-crossing Number

Article

Abstract

The crossing number \({\mbox{\sc cr}}(G)\) of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number \({\mbox{\sc pair-cr}}(G)\) is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number \({\mbox{\sc odd-cr}}(G)\) is the minimum number of pairs of edges that cross an odd number of times. Clearly, \({\mbox{\sc odd-cr}}(G)\le {\mbox{\sc pair-cr}}(G)\le {\mbox{\sc cr}}(G)\) . We construct graphs with \(0.855\cdot {\mbox{\sc pair-cr}}(G)\ge {\mbox{\sc odd-cr}}(G)\) . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte.

Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k2/log 2k).

Keywords

Drawings of a graph Crossing number 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Pach, J., Tóth, G.: Which crossing number is it anyway? J. Comb. Theory Ser. B 80, 225–246 (2000) MATHCrossRefGoogle Scholar
  2. 2.
    Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9, 194–207 (2000) MATHMathSciNetGoogle Scholar
  3. 3.
    Pelsmajer, M., Schaefer, M., Štefankovič, D.: Removing even crossings. In: S. Felsner (ed.) European Conference on Combinatorics, Graph Theory and Applications (EuroComb ’05), DMTCS Conference Volume AE, pp. 105–110 (2005) Google Scholar
  4. 4.
    Pelsmajer, M., Schaefer, M., Štefankovič, D.: Odd crossing number is not crossing number. In: Healy, P., Nikolov, N.S. (eds.) Graph Drawing 2005. Lecture Notes in Computer Science, vol. 3843, pp. 386–396. Springer, Berlin (2006) CrossRefGoogle Scholar
  5. 5.
    Schaefer, M., Štefankovič, D.: Decidability of string graphs. J. Comput. Syst. Sci. 68, 319–334 (2004) MATHCrossRefGoogle Scholar
  6. 6.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Comb. Theory 8, 45–53 (1970) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Valtr, P.: On the pair-crossing number. In: Combinatorial and Computational Geometry. Math. Sci. Res. Inst. Publ., vol. 52, pp. 569–575. Cambridge Univ. Press, Cambridge (2005) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Rényi InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations