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.

20 pages, 12 figures. Fixed a few minor mistakes. Changes suggested by the reviewers have been made. Simplified some formulas

References in corpus (3)

• The chromatic number of the plane is at least 5 -- a human-verifiable proof
• A lower bound on the number of colours needed to nicely colour a sphere
• The surface of a sufficiently large sphere has chromatic number at most 7