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Progress on Crossing Number Problems

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SOFSEM 2005: Theory and Practice of Computer Science (SOFSEM 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3381))

Abstract

Crossing numbers have drawn much attention in the last couple of years and several surveys [22], [28], [33], problem collections [26], [27], and bibliographies [40] have been published. The present survey tries to give pointers to some of the most significant recent developments and identifies computational challenges.

This research was supported in part by the NSF grant DMS 0302307.

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References

  1. Ábrego, B.M., Fernández-Merchant, S.: A lower bound for the rectilinear crossing number (manuscript)

    Google Scholar 

  2. Aigner, M., Ziegler, G.M.: Proofs from the Book. Springer, Berlin (1998)

    MATH  Google Scholar 

  3. Ajtai, N., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. Annals of Discrete Mathematics 12, 9–12 (1982)

    MATH  Google Scholar 

  4. Aichholzer, O., Aurenhammer, F., Krasser, H.: On the crossing number of complete graphs. In: Proc. Ann. ACM Symp. Computational Geometry, Barcelona, Spain, pp. 19–24 (2002)

    Google Scholar 

  5. Aurenhammer, F.: On the Rectilinear Crossing Number, http://www.igi.tugraz.at/auren/

  6. Balogh, J., Salazar, G.: On k-sets, convex quadrilaterals, and the rectilinear crossing number of K n (submitted)

    Google Scholar 

  7. Bienstock, D., Dean, N.: Bounds for rectilinear crossing numbers. J. Graph Theory 17, 333–348 (1991)

    Article  MathSciNet  Google Scholar 

  8. Czabarka, É., Sýkora, O., Székely, L.A.: Vrťo, I.: Convex crossing Numbers, circular arrangement problem, and isoperimetric functions (submitted)

    Google Scholar 

  9. Dey, T.: Improved bounds for planar k-sets and related problems. Discrete Comput. Geom. 19, 373–382 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  10. Elekes, G.: On the number of sums and products. Acta Arithm. 81(4), 365–367 (1997)

    MATH  MathSciNet  Google Scholar 

  11. Even, G., Guha, S., Schieber, B.: Improved approximations of crossings in graph drawings and VLSI layout areas. In: Proc. 32nd Annual Symposium on Theory of Computing, STOC 2000, pp. 296–305. ACM Press, New York (2000)

    Chapter  Google Scholar 

  12. Erdős, P.: On sets of distances of n points. Amer. Math. Monthly 53, 248–250 (1946)

    Article  MathSciNet  Google Scholar 

  13. Fáry, I.: On straight line representations of graphs. Acta Univ. Szeged Sect. Sci. Math. 11, 229–233 (1948)

    MathSciNet  Google Scholar 

  14. Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Alg. Discrete Methods 4, 312–316 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  15. Grohe, M.: Computing crossing numbers in quadratic time. In: Proc. 32nd Annual ACM Symposium on the Theory of Computing, STOC 2001, pp. 231–236 (2001)

    Google Scholar 

  16. Iosevich, A.: Fourier analysis and geometric combinatorics (to appear)

    Google Scholar 

  17. Leighton, F.T.: Complexity Issues in VLSI. MIT Press, Cambridge (1983)

    Google Scholar 

  18. Leighton, F.T., Rao, S.: An approximate max flow min cut theorem for multicommodity flow problem with applications to approximation algorithm. In: Proc. 29th Annual IEEE Symposium on Foundations of Computer Science, pp. 422–431. IEEE Computer Society Press, Washington (1988); J. ACM 46, 787–832 (1999)

    Google Scholar 

  19. 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; Amer. Math. Soc., 139–148 (2004)

    Google Scholar 

  20. Owens, A.: On the biplanar crossing number. IEEE Transactions on Circuit Theory CT-18, 277–280 (1971)

    Article  Google Scholar 

  21. Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley and Sons, New York (1995)

    MATH  Google Scholar 

  22. Pach, J.: Crossing numbers. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 267–273. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  23. Pach, J., Spencer, J., Tóth, G.: New bounds on crossing numbers. In: Proc. 15th ACM Symposium on Computational Geometry, pp. 124–133. ACM, New York (1999); Discrete Comp. Geom. 24, 623–644 (2000)

    Google Scholar 

  24. Pach, J., Tóth, G.: Which crossing number is it anyway? In: Proc. 39th Annual Symposium on Foundation of Computer Science, pp. 617–626. IEEE Press, Baltimore (1998); J. Comb. Theory, Ser. B 80, 225–246 (2000)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  27. Richter, R.B., Salazar, G.: A survey of good crossing number theorems and questions (manuscript)

    Google Scholar 

  28. Shahrokhi, F., Sýkora, O., Székely, L.A., Vrťo, I.: Crossing numbers: bounds and applications. In: Bárány, I., Böröczky, K. (eds.) Intuitive Geometry. Bolyai Society Mathematical Studies, vol. 6, pp. 179–206. János Bolyai Mathematical Society, Budapest (1997)

    Google Scholar 

  29. Shahrokhi, F., Sýkora, O., Székely, L.A., Vrťo, I.: The gap between the crossing number and the convex crossing number. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342; Amer. Math. Soc., 249–258 (2004)

    Google Scholar 

  30. Spencer, J., Szemerédi, E., Trotter, W.T.: Unit distances in the Euclidean plane. In: Bollobás, B. (ed.) Graph Theory and Combinatorics, pp. 293–308. Academic Press, London (1984)

    Google Scholar 

  31. Székely, L.A.: Crossing numbers and hard Erdős problems in discrete geometry. Combinatorics, Probability and Computing 6, 353–358 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Székely, L.A.: Zarankiewicz crossing number conjecture. In: Hazewinkel, M. (supplement III Managing ed.) Kluwer Encyclopaedia of Mathematics, pp. 451–452. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  33. Székely, L.A.: A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Math. 276, 1–3, 331–352 (2003)

    Google Scholar 

  34. Székely, L.A.: Short proof for a theorem of Pach, Spencer, and Tóth. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342; Amer. Math. Soc., 281–283 (2004)

    Google Scholar 

  35. Székely, L.A.: An optimality criterion for the crossing number (submitted)

    Google Scholar 

  36. Szemerédi, E., Trotter, W.T.: Extremal problems in discrete geometry. Combinatorica 3, 381–392 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  37. Tóth, C.D.: The Szemerédi-Trotter theorem in the complex plane (manuscript)

    Google Scholar 

  38. Turán, P.: A note of welcome. J. Graph Theory 1, 7–9 (1977)

    Article  Google Scholar 

  39. Tutte, W.T.: Toward a theory of crossing numbers. J. Combinatorial Theory 8, 45–53 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  40. Vrťo, I.: Crossing Numbers of Graphs: A Bibliography, http://sun.ifi.savba.sk/~imrich/

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Authors and Affiliations

  1. University of South Carolina, Columbia, SC, 29208, USA

    László A. Székely

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  1. László A. Székely
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Editors and Affiliations

  1. Charles University, Prague, Czech Republic

    Peter Vojtáš

  2. Institute of Informatics and Software Engineering Faculty of Informatics and Information technologies, Slovak University of Technology, Ilkovičova 3, 842 16, Bratislava,  

    Mária Bieliková

  3. STIX, École Polytechnique, 91128, Palaiseau Cedex, France

    Bernadette Charron-Bost

  4. Department of Computer Science, University of Loughborough, Loughborough, UK

    Ondrej Sýkora

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Székely, L.A. (2005). Progress on Crossing Number Problems. In: Vojtáš, P., Bieliková, M., Charron-Bost, B., Sýkora, O. (eds) SOFSEM 2005: Theory and Practice of Computer Science. SOFSEM 2005. Lecture Notes in Computer Science, vol 3381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30577-4_8

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  • DOI: https://doi.org/10.1007/978-3-540-30577-4_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24302-1

  • Online ISBN: 978-3-540-30577-4

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