On the edge-vertex ratio of maximal thrackles

TL;DR

This paper explores the properties of maximal thrackles, revealing that their edge-vertex ratio can vary widely, and provides specific bounds and examples for geometric and topological cases.

Contribution

It introduces new bounds and constructions for the edge-vertex ratio of maximal thrackles, advancing understanding of Conway's thrackle conjecture.

Findings

Edge-vertex ratio can be arbitrarily small in maximal thrackles.

Forbidding isolated vertices, the ratio approaches 1/2.

An infinite family of topological thrackles with ratio 5/6.

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.