Abstract
A topological graph G is a graph drawn in the plane so that its edges are represented by Jordan arcs. G is called simple, if any two edges have at most one point in common. It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k − 8 n) edges. The assumption that G is simple cannot be dropped: for every n, there exists a complete topological graph of n vertices, whose any two edges cross at most twice.
János Pach has been supported by NSF Grant CCR-00-98246, by PSC-CUNY Research Award 65392-0034, OTKA T-030012, and by OTKA T-032452. Géza Tóth has been supported by OTKA T-030012 and OTKA T-038397.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17, 1–9 (1997)
Alon, N., Erdõs, P.: Disjoint edges in geometric graphs. Discrete Comput. Geom. 4, 287–290 (1989)
Chojnacki, C., Hanani, A.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fund. Math. 23, 135–142 (1934)
Kolman, P., Matoušek, J.: Crossing number, pair-crossing number, and expansion. Journal of Combinatorial Theory, Ser. B (to appear)
Leighton, F.T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. Assoc. Comput. Machin. 46, 787–832 (1999)
Pach, J.: Geometric graph theory. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Notes, vol. 267, pp. 167–200. Cambridge University Press, Cambridge (1999)
Pach, J., Radoičić, R., Tóth, G.: On quasi-planar graphs. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, AMS, vol. 342 (to appear)
Pach, J., Radoičić, R., Tóth, G.: Relaxing planarity for topological graphs. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 221–232. Springer, Heidelberg (2003)
Pach, J., Shahrokhi, F., Szegedy, M.: Applications of the crossing number. Algorithmica 16, 111–117 (1996)
Pach, J., Solymosi, J., Tóth, G.: Unavoidable configurations in complete topological graphs. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 328–337. Springer, Heidelberg (2001); Also in: Discrete and Computational Geometry (accepted)
Pach, J., Tóth, G.: Which crossing number is it anyway? Journal of Combinatorial Theory, Series B 80, 225–246 (2000)
Pach, J., Törõcsik, J.: Some geometric applications of Dilworth’s theorem. Discrete and Computational Geometry 12, 1–7 (1994)
Raghavan, P., Thompson, C.D.: Randomized rounding: A technique for provably good algorithms and algorithmic proof. Combinatorica 7, 365–374 (1987)
Tóth, G.: Note on geometric graphs. J. Combin. Theory, Ser. A 89, 126–132 (2000)
Valtr, P.: On geometric graphs with no k pairwise parallel edges. Discrete and Computational Geometry 19, 461–469 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pach, J., Tóth, G. (2005). Disjoint Edges in Topological Graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds) Combinatorial Geometry and Graph Theory. IJCCGGT 2003. Lecture Notes in Computer Science, vol 3330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30540-8_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-30540-8_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24401-1
Online ISBN: 978-3-540-30540-8
eBook Packages: Computer ScienceComputer Science (R0)