Skip to main content
Springer Nature Link
Log in

Unavoidable Patterns in Complete Simple Topological Graphs

  • Conference paper
  • First Online:
Graph Drawing and Network Visualization (GD 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13764))

Included in the following conference series:

  • 604 Accesses

Abstract

We show that every complete n-vertex simple topological graph contains a topological subgraph on at least \((\log n)^{1/4 - o(1)}\) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound \(\varOmega (\log ^{1/8}n)\) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least \((\log n)^{1 -o(1)}\).

Supported by NSF CAREER award DMS-1800746 and NSF award DMS-1952786.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ábrego, B., et al.: All good drawings of small complete graphs. In: Proceedings 31st European Workshop on Computational Geometry, pp. 57–60 (2015)

    Google Scholar 

  2. Aichholzer, O., García, A., Tejel, J., Vogtenhuber, B., Weinberger, A.: Twisted ways to find plane structures in simple drawings of complete graphs, In Proceedings 38th Symposium Computational Geometry, LIPIcs, Dagstuhl, Germany, pp. 1–18 (2022)

    Google Scholar 

  3. Alon, N., Spencer, J.: The Probabilistic Method, 4th edn. John Wiley & Sons Inc., Hoboken, New Jersey (2016)

    Google Scholar 

  4. Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer-Verlag, Berlin, Germany (2005). https://doi.org/10.1007/0-387-29929-7

  5. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math. 51, 161–166 (1950)

    Article  MathSciNet  Google Scholar 

  6. Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)

    MathSciNet  Google Scholar 

  7. Fox, J., Pach, J., Sudakov, B., Suk, A.: Erdős-Szekeres-type theorems for monotone paths and convex bodies. Proc. Lond. Math. Soc. 105, 953–982 (2012)

    Article  MathSciNet  Google Scholar 

  8. Fulek, R., Ruiz-Vargas, A.: Topological graphs: empty triangles and disjoint matchings. In: Proceedings 29th Symposium Computational Geometry, pp. 259–265. ACM Press, New York (2013)

    Google Scholar 

  9. Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Am. Math. Mon. 98, 165–166 (1991)

    Article  Google Scholar 

  10. Harborth, H., Mengersen, I.: Drawings of the complete graph with maximum number of crossings. Congr. Numer. 88, 225–228 (1992)

    MathSciNet  Google Scholar 

  11. Kynčl, J.: Improved eumeration of simple topological graphs. Disc. Comput. Geom. 50, 727–770 (2013)

    Article  Google Scholar 

  12. Kynčl, J., Valtr, P.: On edges crossing few other edges in simple topological complete graphs. Discret. Math. 309, 1917–1923 (2009)

    Article  MathSciNet  Google Scholar 

  13. Pach, J.: Geometric graph theory. In: Goodman, J., O’Rourke, J., Tóth, C. (eds.) Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, Florida, pp. 257–279 (2017)

    Google Scholar 

  14. Pach, J., Solymosi, J., Tóth, G.: Unavoidable configurations in complete topological graphs. Disc. Comput. Geom. 30, 311–320 (2003)

    Article  MathSciNet  Google Scholar 

  15. Pach, J., Tóth, G.: Disjoint edges in topological graphs. In: Akiyama, J., Baskoro, E.T., Kano, M. (eds.) IJCCGGT 2003. LNCS, vol. 3330, pp. 133–140. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30540-8_15

    Chapter  Google Scholar 

  16. Ruiz-Vargas, A.: Many disjoint edges in topological graphs. Comput. Geom. 62, 1–13 (2017)

    Article  MathSciNet  Google Scholar 

  17. Ruiz-Vargas, A., Suk, A., Tóth, C.: Disjoint edges in topological graphs and the tangled-thrackle conjecture. European J. Combin. 51, 398–406 (2016)

    Article  MathSciNet  Google Scholar 

  18. Scheucher, M.: Personal communication

    Google Scholar 

  19. Suk, A.: Disjoint edges in complete topological graphs. Disc. Comput. Geom. 49, 280–286 (2013)

    Article  MathSciNet  Google Scholar 

  20. Suk, A.: On the Erdos-Szekeres convex polygon problem. J. Amer. Math. Soc. 30, 1047–1053 (2017)

    Article  MathSciNet  Google Scholar 

  21. Suk, A., Walczak, B.: New bounds on the maximum number of edges in \(k\)-quasi-planar graphs. Comput. Geom. 50, 24–33 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    Andrew Suk & Ji Zeng

Authors
  1. Andrew Suk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji Zeng .

Editor information

Editors and Affiliations

  1. John Cabot University, Rome, Italy

    Patrizio Angelini

  2. Kiel University, Kiel, Germany

    Reinhard von Hanxleden

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Suk, A., Zeng, J. (2023). Unavoidable Patterns in Complete Simple Topological Graphs. In: Angelini, P., von Hanxleden, R. (eds) Graph Drawing and Network Visualization. GD 2022. Lecture Notes in Computer Science, vol 13764. Springer, Cham. https://doi.org/10.1007/978-3-031-22203-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22203-0_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22202-3

  • Online ISBN: 978-3-031-22203-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics