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Unavoidable Patterns and Plane Paths in Dense Topological Graphs

Authors Balázs Keszegh , Andrew Suk , Gábor Tardos , Ji Zeng

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LIPIcs.SoCG.2026.63.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
  • Mathematics of computing → Graphs and surfaces
Keywords
  • graph drawing
  • topological graph
  • bipartite geometric graph
  • forbidden subgraph
  • extremal graph
  • thrackle

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Abstract

Let C_{s,t} be the complete bipartite geometric graph, with s and t vertices on two distinct parallel lines respectively, and all s t straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size 2(k-1)⁴ + 1 and 2^{k^{5k}}, contains a topological subgraph weakly isomorphic to C_{k,k}. As a corollary, every n-vertex simple topological graph not containing a plane path of length k has at most O_k(n^{2 - 8/k⁴}) edges. When k = 3, we obtain a stronger bound by showing that every n-vertex simple topological graph not containing a plane path of length 3 has at most O(n^{4/3}) edges. We also prove that x-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.

Cite As Get BibTex

Balázs Keszegh, Andrew Suk, Gábor Tardos, and Ji Zeng. Unavoidable Patterns and Plane Paths in Dense Topological Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.63

Author Details

Balázs Keszegh
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • ELTE Eötvös Loránd University, Budapest, Hungary
Andrew Suk
  • Department of Mathematics, University of California at San Diego, La Jolla, CA, USA
Gábor Tardos
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Ji Zeng
  • HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

Funding

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930, while the authors were in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the 2025 Extremal Combinatorics Program.
  • Keszegh, Balázs: Supported by the ERC Advanced Grant "ERMiD", no. 101054936 and by the EXCELLENCE-24 project no. 151504 Combinatorics and Geometry of the NRDI Fund.
  • Suk, Andrew: Supported by NSF grant DMS-2246847.
  • Tardos, Gábor: Supported by ERC Advanced Grants "GeoScape", no. 882971 and "ERMiD", no. 101054936.
  • Zeng, Ji: Supported by ERC Advanced Grants "GeoScape", no. 882971 and "ERMiD", no. 101054936.

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