A Computational Approach to Conway’s Thrackle Conjecture

  • Radoslav Fulek
  • János Pach
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)


A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n) = n for every n ≥ 3. For any ε> 0, we give an algorithm terminating in \(e^{O((1/\varepsilon^2)\ln(1/\varepsilon))}\) steps to decide whether t(n) ≤ (1 + ε)n for all n ≥ 3. Using this approach, we improve the best known upper bound, \(t(n)\le \frac 32(n-1)\), due to Cairns and Nikolayevsky, to \(\frac{167}{117}n<1.428n\).


  1. 1.
    Bollobás, B.: Modern Graph Theory. Springer, New York (1998)MATHGoogle Scholar
  2. 2.
    Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)MATHGoogle Scholar
  3. 3.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23, 191–206 (2000)MATHMathSciNetGoogle Scholar
  4. 4.
    Cairns, G., McIntyre, M., Nikolayevsky, Y.: The thrackle conjecture for \(K\sb 5\) and \(K\sb {3,3}\). In: Towards a theory of Geometric Graphs, Contemp. Math. Amer. Math. Soc., vol. 342, pp. 35–54. RI, Providence (2004)Google Scholar
  5. 5.
    Cairns, G., Nikolayevsky, Y.: Generalized thrackle drawings of non-bipartite graphs. Discrete Comput. Geom. 41, 119–134 (2009)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Diestel, R.: Graph Theory. Springer, New York (2008)Google Scholar
  7. 7.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: Trémaux Trees and Planarity. Internat. J. Found. of Comput. Sc. 17, 1017–1030Google Scholar
  8. 8.
    de Fraysseix, H., de Mendez, P.O.: Public Implementation of a Graph Algorithm Library and Editor, http://pigale.sourceforge.net/
  9. 9.
    Green, J.E., Ringeisen, R.D.: Combinatorial drawings and thrackle surfaces. In: Graph Theory, Combinatorics, and Algorithms, Kalamazoo, MI, vol. 2, pp. 999–1009. Wiley-Intersci. Publ., New York (1995)Google Scholar
  10. 10.
    Hopf, H., Pannwitz, E.: Aufgabe Nr. 167. Jahresbericht Deutsch. Math.-Verein. 43, 114 (1934)Google Scholar
  11. 11.
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. Journal of the Association for Computing Machinery 21(4), 549–568Google Scholar
  12. 12.
    Lovász, L., Pach, J., Szegedy, M.: On Conway’s thrackle conjecture. Discrete Comput. Geom. 18, 369–376 (1998)CrossRefGoogle Scholar
  13. 13.
    Perlstein, A., Pinchasi, R.: Generalized thrackles and geometric graphs in R 3 with no pair of strongly avoiding edges. Graphs Combin. 24, 373–389 (2008)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Piazza, B.L., Ringeisen, R.D., Stueckle, S.K.: Subthrackleable graphs and four cycles. Graph theory and Applications (Hakone, 1990), Discrete Math 127, 265–276 (1994)MATHMathSciNetGoogle Scholar
  15. 15.
    Ringeisen, R.D.: Two old extremal graph drawing conjectures: progress and perspectives. Congressus Numerantium 115, 91–103 (1996)MATHMathSciNetGoogle Scholar
  16. 16.
    Rosenstiehl, P.: Solution algebrique du probleme de Gauss sur la permutation des points d’intersection d’une ou plusieurs courbes fermees du plan. C. R. Acad. Sci. Paris Ser. A-B 283, A551–A553 (1976)Google Scholar
  17. 17.
    Sutherland, J.W.: Lösung der Aufgabe 167. Jahresbericht Deutsch. Math.-Verein. 45, 33–35 (1935)Google Scholar
  18. 18.
    Woodall, D.R.: Thrackles and deadlock. In: Welsh, D.J.A. (ed.) Combinatorial Mathematics and Its Applications, vol. 348, pp. 335–348. Academic Press, London (1969)Google Scholar
  19. 19.
    Unsolved problems. Chairman: P. Erdős, in: Combinatorics Proc. Conf. Combinatorial Math., Math. Inst., Oxford, Inst. Math. Appl., Southend-on-Sea, 351–363 (1972)Google Scholar
  20. 20.

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Radoslav Fulek
    • 1
  • János Pach
    • 2
  1. 1.Ecole Polytechnique Fédérale de LausanneSwitzerland
  2. 2.Ecole Polytechnique Fédérale de Lausanne and City CollegeNew York

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