Discrete & Computational Geometry

, Volume 23, Issue 2, pp 191–206 | Cite as

Bounds for Generalized Thrackles

  • G. Cairns
  • Y. Nikolayevsky


A thrackle (resp. generalized thrackle) is a drawing of a graph in which each pair of edges meets precisely once (resp. an odd number of times). For a graph with n vertices and m edges, we show that, for drawings in the plane, m≤ (2/3)(n-1) for thrackles, while m≤ 2n-2 for generalized thrackles. This improves theorems of Lovász, Pach, and Szegedy. The paper also examines thrackles in the more general setting of drawings on closed surfaces. The main result is: a bipartite graph G can be drawn as a generalized thrackle on a closed orientable connected surface if and only if G can be embedded in that surface.


General Setting Bipartite Graph Closed Surface Connected Surface Orientable Connected Surface 
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Copyright information

© Springer-Verlag New York Inc. 2000

Authors and Affiliations

  • G. Cairns
    • 1
  • Y. Nikolayevsky
    • 1
  1. 1.Department of Mathematics, La Trobe University, Melbourne, Victoria 3083, Australia G., Y.

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