Expected Crossing Numbers

Part of the Fields Institute Communications book series (FIC, volume 69)


The expected value for the weighted crossing number of a randomly weighted graph is studied. We focus on the case where G = K n and the edge-weights are independent random variables that are uniformly distributed on [0,1]. The first non-trivial case is K 5. We compute this via an unexpectedly involved calculation, and consider bounds for larger values of n. A variation of the Crossing Lemma for expectations is proved.

Key words

Graph Crossing number Weighted crossing number Crossing lemma 

Subject Classifications

05C10 60C05 



An extended abstract containing preliminary versions of these results appeared in the proceedings of EUROCOMB 2011 [9].

B. Mohar was supported in part by an NSERC Discovery Grant (Canada), by the Canada Research Chair program, and by the Research Grant P1–0297 of ARRS (Slovenia). He is on leave from: IMFM & FMF, Department of Mathematics, University of Ljubljana, Ljubljana, Slovenia. T. Stephen was supported in part by an NSERC Discovery Grant. The authors are grateful to Luis Goddyn for some helpful discussions on the subject, and to the anonymous referees for helpful comments.


  1. 1.
    Aichholzer, O., Aurenhammer, F., Krasser, H.: On the crossing number of complete graphs. Computing 76, 165–176 (2006). doi:10.1007/s00607-005-0133-3MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. In: Turgeon, J., Rosa, A., Sabidussi, G. (eds.) Theory and Practice of Combinatorics. North-Holland Mathematics Studies, vol. 60, pp. 9–12. North-Holland, Amsterdam (1982)Google Scholar
  3. 3.
    David, H.A., Nagaraja, H.N.: Order Statistics. Wiley Series in Probability and Statistics, 3rd edn. Wiley-Interscience, Hoboken (2003)Google Scholar
  4. 4.
    Guy, R.K.: Crossing numbers of graphs. In: Graph Theory and Applications. Proceedings of the Conference, Western Michigan University, Kalamazoo, 1972, pp. 111–124. Lecture Notes in Mathematics, vol. 303. Springer, Berlin (1972)Google Scholar
  5. 5.
    Leighton, F.T.: New lower bound techniques for VLSI. Math. Syst. Theory 17, 47–70 (1984). doi:10.1007/BF01744433MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Lovász, L., Vesztergombi, K., Wagner, U., Welzl, E.: Convex quadrilaterals and k-sets. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 139–148. American Mathematical Society, Providence (2004)CrossRefGoogle Scholar
  7. 7.
    Mohar, B.: Crossing numbers of graphs on the plane and on other surfaces. In: Abstracts of the 20th Workshop on Topological Graph Theory in Yokohama, 25–28 Nov 2008, Yokohama (2008)Google Scholar
  8. 8.
    Mohar, B.: Do we really understand the crossing numbers? In: Hliněný, P., et al. (eds.) Mathematical Foundations of Computer Science 2010: Proceedings of the 35th International Symposium, MFCS 2010, Brno, 23–27 Aug 2010. Lecture Notes in Computer Science, vol. 6281, pp. 38–41. Springer, Berlin (2010)Google Scholar
  9. 9.
    Mohar, B., Stephen, T.: Expected crossing numbers. In: Nešetřil, J., Györi, E., Sali, A. (eds.) Proceedings of the European Conference on Combinatorics, Graph Theory and Applications, Budapest. Electronic Notes in Discrete Mathematics, vol. 38, pp. 651–656. Elsevier, Oxford (2011)Google Scholar
  10. 10.
    Pach, J., Tóth, G.: Which crossing number is it anyway? J. Comb. Theory Ser. B 80(2), 225–246 (2000)zbMATHCrossRefGoogle Scholar
  11. 11.
    Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the crossing lemma by finding more crossings in sparse graphs. Discret. Comput. Geom. 36, 527–552 (2006). doi:10.1007/s00454-006-1264-9zbMATHCrossRefGoogle Scholar
  12. 12.
    Pan, S., Richter, R.B.: The crossing number of K 11 is 100. J. Graph Theory 56, 128–134 (2007). doi:10.1002/jgt.20249MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Richter, R.B., Salazar, G.: Crossing numbers. In: Beineke, L.W., Wilson, R.J. (eds.) Topics in Topological Graph Theory. Encyclopedia of Mathematics and Its Applications, vol. 128, pp. 133–150. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  14. 14.
    Shahrokhi, F., Székely, L., Vrt’o, I.: Crossing numbers of graphs, lower bound techniques and algorithms: a survey. In: Tamassia, R., Tollis, I. (eds.) Graph Drawing. Lecture Notes in Computer Science, vol. 894, pp. 131–142. Springer, Berlin/New York (1995).
  15. 15.
    Spencer, J., Tóth, G.: Crossing numbers of random graphs. Random Struct. Algorithms 21(3–4), 347–358 (2002)zbMATHCrossRefGoogle Scholar
  16. 16.
    Vrt’o, I.: Crossing numbers of graphs: a bibliography (2010). Available at:

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.CRC in Graph Theory, Tier 1, Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada

Personalised recommendations