Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres
Abstract
This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Nelson-Hadwiger problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not contain the Moser spindle as a subgraph. The construction of 5-chromatic graphs embedded in a sphere at two values of the radius is given. Namely, the 5-chromatic unit distance graph on 372 vertices embedded into the circumsphere of an icosahedron with a unit edge length, and the 5-chromatic graph on 972 vertices embedded into the circumsphere of a great icosahedron are constructed.
Cite
@article{arxiv.2106.11824,
title = {Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres},
author = {Vsevolod Voronov and Anna Neopryatnaya and Eugene Dergachev},
journal= {arXiv preprint arXiv:2106.11824},
year = {2022}
}
Comments
20 pages, 12 figures. Fixed a few minor mistakes. Changes suggested by the reviewers have been made. Simplified some formulas