# Thrackles: An Improved Upper Bound

@inproceedings{Fulek2017ThracklesAI,
title={Thrackles: An Improved Upper Bound},
author={Radoslav Fulek and J{\'a}nos Pach},
booktitle={Graph Drawing},
year={2017}
}
• Published in Graph Drawing 27 August 2017
• Mathematics
A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is $${3\over 2}(n-1)$$, and that this bound is best…
3 Citations
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The Thrackle Conjecture is proved for thrackle drawings all of whose vertices lie on the boundaries of connected domains in the complement of the drawing.
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A thrackle is a graph drawing in which every pair of edges meets exactly once. Conway’s Thrackle Conjecture states that the number of edges of a thrackle cannot exceed the number of its vertices.
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The maximum rectilinear crossing number of a graph $G$ is the maximum number of crossings in a good straight-line drawing of $G$ in the plane. In a good drawing any two edges intersect in at most one

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