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 \({\rm cr}(G') \geq (\frac{1}{28}-o(1)){\rm cr}(G)\).
Chapter PDF
References
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)
Erdős, P., Simonovits, M.: Compactness results in extremal graph theory. Combinatorica 2, 275–288 (1982)
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)
Leighton, T.: Complexity Issues in VLSI. MIT Press, Cambridge (1983)
Leighton, T.: New lower bound techniques for VLSI. Math. Systems Theory 17, 47–70 (1984)
Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)
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)
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)
Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. Algorithmica 16, 111–117 (1996)
Pach, J., Solymosi, J., Tardos, G.: Crossing numbers of imbalanced graphs. Lecture presented at SIAM Conf. Discrete Math., Victoria, BC (2006)
Pach, J., Spencer, J., Tóth, G.: New bounds on crossing numbers. Discrete Comput. Geom. 24, 623–644 (2000)
Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9, 194–207 (2000)
Richter, B., Thomassen, C.: Minimal graphs with crossing number at least k. J. Combin. Theory Ser. B 58, 217–224 (1993)
Salazar, G.: On a crossing number result of Richter and Thomassen. J. Combin. Theory Ser. B 79, 98–99 (2000)
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)
Székely, L.A.: A successful concept for measuring non-planarity of graphs: the crossing number. Discrete Math. 276, 331–352 (2004)
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)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Fox, J., Tóth, C.D. (2007). On the Decay of Crossing Numbers. In: Kaufmann, M., Wagner, D. (eds) Graph Drawing. GD 2006. Lecture Notes in Computer Science, vol 4372. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70904-6_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-70904-6_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70903-9
Online ISBN: 978-3-540-70904-6
eBook Packages: Computer ScienceComputer Science (R0)