Great-circle Tree Thrackles

TL;DR

This paper investigates great-circle thrackles, a special class of thrackle drawings on the sphere, revealing that not all trees can be represented as such, thus challenging a previous conjecture relating these drawings to the general thrackle conjecture.

Contribution

It demonstrates that the class of graphs drawable as great-circle thrackles does not include all trees, disproving the equivalence of the conjecture to the general thrackle conjecture.

Findings

Great-circle thrackleable graphs exclude some trees.

The informal conjecture by Cairns, Koussas, and Nikolayevsky is not equivalent to the Thrackle Conjecture.

The class of great-circle thrackleable graphs is strictly smaller than all graphs.

Abstract

A thrackle is a graph drawing in which every pair of edges meets exactly once. The Thrackle Conjecture (established by John Conway) states that the number of edges of a thrackle cannot exceed the number of its vertices. Cairns, Koussas, and Nikolayevsky (2015) prove that the Thrackle Conjecture holds for great-circle thrackles drawn on the sphere. They also posit that the Thrackle Conjecture can be restated to say that a graph can be drawn as a thrackle drawing in the plane if and only if it admits a great-circle thrackle drawing. We demonstrate that the class of great-circle thrackleable graphs excludes some trees. Thus the informal conjecture from Cairns, Koussas, and Nikolayevsky (2015) is not equivalent to the Thrackle Conjecture.