Bounds for generalized thrackles

A thrackle (resp. generalized thrackle) is a drawing of a graph in which ea ch 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 less than or equal to 3/2 (n - 1) far thrackles, while m less than or e qual to 2n - 2 for generalized thrackles. This improves theorems of Lovasz, Pach, and Szegedy. The paper also examines thrackles in the more general s etting of drawings on closed surfaces. The main result is: a bipartite grap h 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.