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Discrete & Computational Geometry

, Volume 58, Issue 2, pp 410–416 | Cite as

On the Bounds of Conway’s Thrackles

  • Luis Goddyn
  • Yian XuEmail author
Article

Abstract

A thrackle on a surface X is a graph of size e and order n drawn on X such that every two distinct edges of G meet exactly once either at their common endpoint, or at a proper crossing. An unsolved conjecture of Conway (1969) asserts that \(e\le n\) for every thrackle on a sphere. Until now, the best known bound is \(e\le 1.428n\). By using discharging rules we show that \(e\le 1.4n-1.4\).

Keywords

Thrackle Generalized thrackle Crossing number 

Mathematics Subject Classification

05C10 05C62 68R10 

Notes

Acknowledgements

We would like to thank the referees for their great suggestions which helped us a lot in improving the presentation, especially for their suggestions on reducing the upper bound from 1.4n to \(1.4(n-1)\).

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  2. 2.School of Mathematics and StatisticsThe University of Western AustraliaPerthAustralia

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