Upper bound on the circular chromatic number of the plane
type: Item
instance of: scholarly article
title: Upper bound on the circular chromatic number of the plane
zbMATH Open document ID: 1391.05110
author: Konstanty Junosza-Szaniawski
published in: The Electronic Journal of Combinatorics
publication date: 2018-05-25T00:00:00Z
full work available at URL: v25i1p53
review text: Summary: We consider circular version of the famous Nelson-Hadwiger problem. It is know that 4 colors are necessary and 7 colors suffice to color the euclidean plane in such a way that points at distance one get different colors. In \(r\)-circular coloring we assign arcs of length one of a circle with a perimeter \(r\) in such a way that points at distance one get disjoint arcs. In this paper we show the existence of \(r\)-circular coloring for \(r=4+\frac{4\sqrt{3}}{3}\approx 6.30\). It is the first result with \(r\)-circular coloring of the plane with \(r\) smaller than 7. We also show \(r\)-circular coloring of the plane with \(r<7\) in the case when we require disjoint arcs for points at distance belonging to the internal [0.9327,1.0673].
Mathematics Subject Classification ID: 05C15
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C62
zbMATH DE Number: 6873097
zbMATH Keywords: circular colouring
zbMATH Keywords: Hadwiger-Nelson problem
zbMATH Keywords: coloring of the plane
MaRDI profile type: Publication
cites work: Circular Coloring the Plane
cites work: \(\varepsilon\)-unit distance graphs
cites work: Fractional and \(j\)-fold coloring of the plane
cites work: On Harnack's theorem for elliptic differential equations
cites work: Star chromatic number
cites work: Circular chromatic number: A survey
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description: scientific article; zbMATH DE number 6873097