, Volume 23, Issue 2, pp 191-206

Bounds for Generalized Thrackles

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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.

Received July 23, 1998, and in revised form January 1, 1999.