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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 ≤ 3 2 (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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C.F. Gauss gave a necessary condition for a word to be the intersection word of a closed normal planar curve and he gave an example which showed that his condition was not sufficient. M. Dehn provided a solution to the planarity problem [3] and subsequently, different solutions have been given by a number of authors (see [9]). However, all of these solutions are algorithmic in nature. As B. Grünbaum remarked in [7], “they are of the same aesthetically unpleasing character as MacLane’s [1937] criterion for planarity of graphs. A characterization of Gauss codes in the spirit of the Kuratowski criterion for planarity of graphs is still missing”. In this paper we use the work of J. Scott Carter [2] to give a necessary and sufficient condition for planarity of signed Gauss words which is analogous to Gauss’s original condition.