Great-circle Tree Thrackles
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.
Keywords
Cite
@article{arxiv.2110.09679,
title = {Great-circle Tree Thrackles},
author = {Karen Collins and Cleo Roberts},
journal= {arXiv preprint arXiv:2110.09679},
year = {2021}
}
Comments
15 pages, 18 figures