Maximal geometric thrackles without isolated vertices can have edge-vertex ratio arbitrarily close to 1/2, and there is an infinite family of maximal topological thrackles with ratio 5/6.
Discrete & Computational Geometry 23(2), 191–206 (2000)
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On the edge-vertex ratio of maximal thrackles
Maximal geometric thrackles without isolated vertices can have edge-vertex ratio arbitrarily close to 1/2, and there is an infinite family of maximal topological thrackles with ratio 5/6.