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Discrete & Computational Geometry

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

Bounds for Generalized Thrackles

  • G. Cairns
  • Y. Nikolayevsky

Abstract.

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.

Keywords

General Setting Bipartite Graph Closed Surface Connected Surface Orientable Connected Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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. Cairns@latrobe.edu.au, Y. Nikolayevsky@latrobe.edu.auAU

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