Discrete & Computational Geometry

, Volume 18, Issue 4, pp 369–376

On Conway's Thrackle Conjecture

  • L. Lovász
  • J. Pach
  • M. Szegedy

DOI: 10.1007/PL00009322

Cite this article as:
Lovász, L., Pach, J. & Szegedy, M. Discrete Comput Geom (1997) 18: 369. doi:10.1007/PL00009322


A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About 40 years ago, J. H. Conway conjectured that the number of edges of a thrackle cannot exceed the number of its vertices. We show that a thrackle has at most twice as many edges as vertices. Some related problems and generalizations are also considered.

Copyright information

© 1997 Springer-Verlag New York Inc.

Authors and Affiliations

  • L. Lovász
    • 1
  • J. Pach
    • 2
  • M. Szegedy
    • 3
  1. 1. Department of Computer Science, Yale University, New Haven, CT 06517, USA lovasz-laszlo@cs.elte.huUS
  2. 2. Department of Computer Science, City College, CUNY and Courant Institute, NYU, 251 Mercer Street, New York, NY 10012, USA pach@cims.nyu.eduUS
  3. 3. Mathematical Sciences Research Center, AT&T Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA ms@research.att.comUS