Skip to main content
SpringerLink
Log in

Relaxing Planarity for Topological Graphs

  • Chapter
Book cover More Sets, Graphs and Numbers

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 15))

Abstract

According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straight-line drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known.

János Pach has been supported by NSF Grant CCR-00-98245, by PSC-CUNY Research Award 63352-0036, and by OTKA T-032458. Géza Tóth has been supported by OTKA-T-038397 and by an award from the New York University Research Challenge Fund.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. K. Agarwal, B. Aronov, J. Pach, R. Pollack and M. Sharir, Quasi-planar graphs have a linear number of edges, Combinatorica, 17 (1997), 1–9.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Ajtai, V. Chvátal, M. Newborn and E. Szemerédi, Crossing-free subgraphs, in: Theory and Practice of Combinatorics, North-Holland Math. Stud., 60, North-Holland, Amsterdam-New York, 1982, 9–12.

    Google Scholar 

  3. N. Alon and P. Erdős, Disjoint edges in geometric graphs, Discrete Comput. Geom., 4 (1989), 287–290.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bollobás and A. Thomason, Proof of a conjecture of Mader, Erdős and Hajnal on topological complete subgraphs, European J. Combin., 19 (1998), 883–887.

    Article  MATH  MathSciNet  Google Scholar 

  5. P. Braß, G. Károlyi and P. Valtr, A Turán-type extremal theory for convex geometric graphs, in: Discrete and Computational Geometry — The Goodman-Pollack Festschrift (B. Aronov et al., eds.), Springer Verlag, Berlin, 2003, to appear.

    Google Scholar 

  6. G. Cairns and Y. Nikolayevsky, Bounds for generalized thrackles, Discrete Comput. Geom., 23 (2000), 191–206.

    Article  MATH  MathSciNet  Google Scholar 

  7. V. Capoyleas and J. Pach, A Turán-type theorem on chords of a convex polygon, Journal of Combinatorial Theory, Series B, 56 (1992), 9–15.

    Article  MATH  MathSciNet  Google Scholar 

  8. W. Goddard, M. Katchalski and D. J. Kleitman, Forcing disjoint segments in the plane, European J. Combin., 17 (1996), 391–395.

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Hopf and E. Pannwitz, Aufgabe Nr. 167, Jahresbericht der deutschen Mathematiker-Vereinigung, 43 (1934), 114.

    Google Scholar 

  10. J. Komlós and E. Szemerédi, Topological cliques in graphs II, Combin. Probab. Comput., 5 (1996), 79–90.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. V. Kostochka, Lower bound of the Hadwiger number of graphs by their average degree, Combinatorica, 4 (1984), 307–316.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. V. Kostochka, Upper bounds on the chromatic number of graphs (in Russian), Trudy Inst. Mat. (Novosibirsk), Modeli i Metody Optim., 10 (1988), 204–226.

    MathSciNet  Google Scholar 

  13. A. V. Kostochka and J. Kratochvíl, Covering and coloring polygon-circle graphs, Discrete Math., 163 (1997), 299–305.

    Article  MATH  MathSciNet  Google Scholar 

  14. F. T. Leighton, New lower bound techniques for VLSI, Math. Systems Theory, 17 (1984), 47–70.

    Article  MATH  MathSciNet  Google Scholar 

  15. L. Lovász, J. Pach and M. Szegedy, On Conway’s thrackle conjecture, Discrete and Computational Geometry, 18 (1997), 369–376.

    Article  MATH  MathSciNet  Google Scholar 

  16. W. Mader, 3n — 5 edges do force a subdivision of K 5, Combinatorica, 18 (1998), 569–595.

    Article  MATH  MathSciNet  Google Scholar 

  17. P. Kolman and J. Matoušek, Crossing number, pair-crossing number, and expansion, J. Comb. Theory, Ser. B, 92 (2004), 99–113.

    Article  MATH  Google Scholar 

  18. S. McGuinness, Colouring arcwise connected sets in the plane I, Graphs & Combin., 16 (2000), 429–439.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Pach, Geometric graph theory, in: Surveys in Combinatorics, 1999 (J. D. Lamb and D. A. Preece, eds.), London Mathematical Society Lecture Notes, 267, Cambridge University Press, Cambridge, 1999, 167–200.

    Google Scholar 

  20. J. Pach, R. Pinchasi, M. Sharir and G. Tóth, Topological graphs with no large grids, Special Issue dedicated to Victor Neumann-Lara, Graphs and Combinatorics (accepted).

    Google Scholar 

  21. J. Pach, R. Pinchasi, G. Tardos and G. Tóth, Geometric graphs with no self-intersecting path of length three, in: Graph Drawing (M. T. Goodrich, S. G. Kobourov, eds.), Lecture Notes in Computer Science 2528, Springer-Verlag (Berlin, 2002), 295–311. Also in: European Journal of Combinatorics, 25 (2004), 793–811.

    Google Scholar 

  22. J. Pach, R. Radoičić and G. Tóth, A generalization of quasi-planarity, in: Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics 342, AMS (2004), 177–183.

    Google Scholar 

  23. J. Pach, R. Radoičić, G. Tardos and G. Tóth, Improving the Crossing Lemma by finding more crossings in sparse graphs, Proceedings of the 20th Annual Symposium on Computational Geometry (2004), 68–75. Also in: Discrete and Computational Geometry, accepted.

    Google Scholar 

  24. J. Pach, F. Shahrokhi and M. Szegedy, Applications of the crossing number, Algorithmica, 16 (1996), 111–117.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Pach and J. Törőcsik, Some geometric applications of Dilworth’s theorem, Discrete Comput. Geom., 12 (1994), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Pinchasi and R. Radoičić, Topological graphs with no self-intersecting cycle of length 4, Proceedings of the 19th Annual Symposium on Computational Geometry (2003), 98–103. Also in: Towards a Theory of Geometric Graphs (J. Pach, ed.), Contemporary Mathematics 342, AMS (2004), 233–243.

    Google Scholar 

  28. N. Robertson, P. Seymour and R. Thomas, Hadwiger’s conjecture for K 6-free graphs, Combinatorica, 13 (1993), 279–361.

    Article  MATH  MathSciNet  Google Scholar 

  29. O. Sýkora and I. Vrt’o, On VLSI layouts of the star graph and related networks, Integration, The VLSI Journal, 17 (1994), 83–93.

    Article  MATH  Google Scholar 

  30. G. Tardos, Construction of locally plane graphs, manuscript.

    Google Scholar 

  31. A. Thomason, An extremal function for contractions of graphs, Math. Proc. Cambridge Philos. Soc., 95 (1984), 261–265.

    Article  MATH  MathSciNet  Google Scholar 

  32. G. Tóth, Note on geometric graphs, J. Combin. Theory, Ser. A, 89 (2000), 126–132.

    Article  MATH  MathSciNet  Google Scholar 

  33. A. Thomason, The extremal function for complete minors, J. Combin. Theory Ser. B, 81 (2001), 318–338.

    Article  MATH  MathSciNet  Google Scholar 

  34. P. Valtr, On geometric graphs with no k pairwise parallel edges, Discrete and Computational Geometry, 19 (1998), 461–469.

    Article  MATH  MathSciNet  Google Scholar 

  35. D. R. Woodall, Thrackles and deadlock, in: Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), Academic Press, London, 1971, 335–347.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    János Pach

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Radoš Radoičić

  3. Rényi Institute, Hungarian Academy of Sciences, H-1364, Budapest, P.O.B. 127, Hungary

    Géza Tóth

Authors
  1. János Pach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radoš Radoičić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Géza Tóth
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Hungarian Academy of Sciences, Alfréd Rényi Inst. of Mathematics, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    Ervin Győri  & Gyula O. H. Katona  & 

  2. Eötvös University (Budapest)/Microsoft (Seattle), Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    László Lovász

  3. Budapest University of Technology and Economics, Pázmány Péter sétány 1/D, 1117, Budapest, Hungary

    Tamás Fleiner

Rights and permissions

Reprints and permissions

Copyright information

© 2006 János Bolyai Mathematical Society and Springer Verlag

About this chapter

Cite this chapter

Pach, J., Radoičić, R., Tóth, G. (2006). Relaxing Planarity for Topological Graphs. In: Győri, E., Katona, G.O.H., Lovász, L., Fleiner, T. (eds) More Sets, Graphs and Numbers. Bolyai Society Mathematical Studies, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32439-3_12

Download citation

Publish with us

Policies and ethics

Search

Navigation