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arXiv · 1908.08857

On the edge-vertex ratio of maximal thrackles

Oswin Aichholzer·Linda Kleist·Boris Klemz·Felix Schröder·Birgit Vogtenhuber
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Abstract

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'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 1/2. For maximal topological thrackles without isolated vertices, we present an infinite family with an edge-vertex ratio of 5/6.

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Oswin Aichholzer, Linda Kleist, Boris Klemz, Felix Schröder, Birgit Vogtenhuber. 2019-09-17. On the edge-vertex ratio of maximal thrackles. https://arxiv.org/abs/1908.08857

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