On the Edge-Vertex Ratio of Maximal Thrackles

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway\â\\s conjecture states that a thrackle has at most as many edges as vertices. In this paper, we investigate the edge-vertex ratio of maximal thrackles, that is, thrackles in which no edge between already existing vertices can be inserted such that the resulting drawing remains a thrackle. For maximal geometric and topological thrackles, we show that the edge-vertex ratio can be arbitrarily small. When forbidding isolated vertices, the edge-vertex ratio of maximal geometric thrackles can be arbitrarily close to the natural lower bound of $12$. For maximal topological thrackles without isolated vertices, we present an infinite family with an edge-vertex ratio arbitrary close to~$ 45$.