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Unavoidable Patterns in Complete Simple Topological Graphs

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Graph Drawing and Network Visualization (GD 2022)

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

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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.

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Authors and Affiliations

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

    Andrew Suk & Ji Zeng

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  1. Andrew Suk
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  2. Ji Zeng
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Correspondence to Ji Zeng .

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Editors and Affiliations

  1. John Cabot University, Rome, Italy

    Patrizio Angelini

  2. Kiel University, Kiel, Germany

    Reinhard von Hanxleden

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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

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

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22202-3

  • Online ISBN: 978-3-031-22203-0

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