Discrete & Computational Geometry

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

On the Bounds of Conway’s Thrackles

  • Luis Goddyn
  • Yian XuEmail author


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 \(e\le 1.4n-1.4\).


Thrackle Generalized thrackle Crossing number 

Mathematics Subject Classification

05C10 05C62 68R10 



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 \(1.4(n-1)\).


  1. 1.
    Archdeacon, D., Stor, K.: Superthrackles. Australas. J. Comb. 67(2), 145–158 (2017)MathSciNetGoogle Scholar
  2. 2.
    Bondy, J.A., Murty, U.S.R.: Graph Theory. Graduate Texts in Mathematics, vol. 244. Springer, New York (2008)zbMATHGoogle Scholar
  3. 3.
    Cairns, G., Nikolayevsky, Yu.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cairns, G., Nikolayevsky, Yu.: Generalized thrackle drawings of non-bipartite graphs. Discrete Comput. Geom. 41(1), 119–134 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cairns, G., Nikolayevsky, Yu.: Outerplanar thrackles. Graphs Combin. 28(1), 85–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fulek, R., Pach, J.: A computational approach to Conway’s thrackle conjecture. Comput. Geom. 44(6–7), 345–355 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18(4), 369–376 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (2001)Google Scholar
  9. 9.
    Woodall, D.R.: Thrackles and Deadlock. In: Walsh, D.J.A. (ed.) Combinatorial Mathematics and Its Applications, pp. 335–347. Academic Press, London (1971)Google Scholar

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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