Constructing 5-chromatic unit distance graphs embedded in the Euclidean plane and two-dimensional spheres

TL;DR

This paper develops algorithms to construct 5-chromatic unit distance graphs embedded in the plane and spheres, providing new examples that advance understanding of the chromatic number of geometric graphs.

Contribution

It introduces novel algorithms for constructing 5-chromatic unit distance graphs without Moser spindles, including large graphs embedded in the plane and spheres.

Findings

Constructed a 5-chromatic graph with 64513 vertices in the plane.

Created 5-chromatic graphs on 372 and 972 vertices embedded in sphere circumspheres.

Graphs do not contain the Moser spindle as a subgraph.

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.