Discrete & Computational Geometry

, Volume 41, Issue 1, pp 119–134 | Cite as

Generalized Thrackle Drawings of Non-bipartite Graphs

  • Grant CairnsEmail author
  • Yury Nikolayevsky


A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.


Graph drawing Thrackle 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsLa Trobe UniversityMelbourneAustralia

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