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ORTHOGONAL COLORINGS OF THE SPHERE

Part of: Graph theory Discrete geometry

Published online by Cambridge University Press:  04 February 2016

Andreas F. Holmsen  and
Seunghun Lee
Andreas F. Holmsen
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon 305–701, South Korea email andreash@kaist.edu
Seunghun Lee
Affiliation:
Department of Mathematical Sciences, KAIST, Daejeon 305–701, South Korea email prosolver@kaist.ac.kr
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Abstract

An orthogonal coloring of the two-dimensional unit sphere $\mathbb{S}^{2}$, is a partition of $\mathbb{S}^{2}$ into parts such that no part contains a pair of orthogonal points: that is, a pair of points at spherical distance ${\it\pi}/2$ apart. It is a well-known result that an orthogonal coloring of $\mathbb{S}^{2}$ requires at least four parts, and orthogonal colorings with exactly four parts can easily be constructed from a regular octahedron centered at the origin. An intriguing question is whether or not every orthogonal 4-coloring of $\mathbb{S}^{2}$ is such an octahedral coloring. In this paper, we address this question and show that if every color class has a non-empty interior, then the coloring is octahedral. Some related results are also given.

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs 52C10: Erdős problems and related topics of discrete geometry
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 492 - 501
Copyright
Copyright © University College London 2016 

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