Geodesic
Upper bound on the circular chromatic number of the plane
Konstanty Junosza-Szaniawski1
1Warsaw University of Technology, Faculty of Mathematics and Information Science
The electronic journal of combinatorics, Tome 25 (2018) no. 1
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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].
DOI : 10.37236/5418
Classification : 05C15, 05C10, 05C62
Mots-clés : circular colouring, Hadwiger-Nelson problem, coloring of the plane

Konstanty Junosza-Szaniawski  1

1 Warsaw University of Technology, Faculty of Mathematics and Information Science 
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Konstanty Junosza-Szaniawski. Upper bound on the circular chromatic number of the plane. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/5418

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