Work

A Lower Bound on the Number of Colours Needed to Nicely Colour a Sphere

Year: 2025

Type: article

Abstract: The Hadwiger–Nelson problem is about determining the chromatic number of the plane (CNP), defined as the minimum number of colours needed to colour the plane so that no two points of distance 1 have the same colour. In this paper we investigate a related problem for spheres and we use a few natural restrictions on the colouring. Thomassen showed that with these restrictions, the chromatic number of all manifolds satisfying certain properties (including the plane and all spheres with a large enough radius) is at least 7. We prove that with these restrictions, the chromatic number of any sphere with a large enough radius is at least 8. This also gives a new lower bound for the minimum colours needed for colouring the 3-dimensional space with the same restrictions.

Source: Discrete & Computational Geometry

DOI: 10.1007/s00454-025-00736-3

Author: Péter Ágoston

Institutions: Eötvös Loránd University, HUN-REN Alfréd Rényi Institute of Mathematics

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: European Commission, Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, National Research, Development and Innovation Office