A computational approach to Conway's thrackle conjecture

TL;DR

This paper introduces an algorithmic method to analyze Conway's thrackle conjecture, improving the upper bound on the maximum edges in a thrackle, and providing computational tools to approach the conjecture.

Contribution

The paper presents a new algorithmic approach to evaluate the thrackle conjecture and improves the known upper bound on t(n) from 3/2(n-1) to approximately 1.428n.

Findings

Algorithm terminates in subexponential time for given epsilon.

Improved upper bound on t(n) from 1.5n to approximately 1.428n.

Provides computational evidence towards the conjecture.

Abstract

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to decide whether t(n)<(1+eps)n for all n>2. Using this approach, we improve the best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to 167/117n<1.428n.