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Generalized thrackle drawings of non-bipartite graphs. (English) Zbl 1191.05032
Summary: 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 $$\chi (M) - 1$$. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C62 Graph representations (geometric and intersection representations, etc.) 68R10 Graph theory (including graph drawing) in computer science
##### Keywords:
graph drawing; thrackle
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##### References:
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