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.
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Received July 23, 1998, and in revised form January 1, 1999.
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Cairns, G., Nikolayevsky, Y. Bounds for Generalized Thrackles. Discrete Comput Geom 23, 191–206 (2000). https://doi.org/10.1007/PL00009495
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DOI: https://doi.org/10.1007/PL00009495