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On Short Edges in Complete Topological Graphs.

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Title: On Short Edges in Complete Topological Graphs.
Authors: Suk, Andrew1 (AUTHOR) asuk@ucsd.edu
Source: Discrete & Computational Geometry. Jan2026, Vol. 75 Issue 1, p193-204. 12p.
Subjects: Topological graph theory, Complete graphs, Graph theory, Combinatorial geometry, Mathematicians, Matching theory
Abstract: Let h(n) be the minimum integer such that every complete n-vertex simple topological graph contains an edge that crosses at most h(n) other edges. In 2009, Kynčl and Valtr showed that h (n) = O (n 2 / log 1 / 4 n) , and in the other direction, gave constructions showing that h (n) = Ω (n 3 / 2) . In this paper, we prove that h (n) = O (n 7 / 4) . Along the way, we establish a new variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest. [ABSTRACT FROM AUTHOR]
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Abstract:Let h(n) be the minimum integer such that every complete n-vertex simple topological graph contains an edge that crosses at most h(n) other edges. In 2009, Kynčl and Valtr showed that h (n) = O (n 2 / log 1 / 4 n) , and in the other direction, gave constructions showing that h (n) = Ω (n 3 / 2) . In this paper, we prove that h (n) = O (n 7 / 4) . Along the way, we establish a new variant of Chazelle and Welzl's matching theorem for set systems with bounded VC-dimension, which we believe to be of independent interest. [ABSTRACT FROM AUTHOR]
ISSN:01795376
DOI:10.1007/s00454-024-00692-4