
Konstanty JunoszaSzaniawski
Keywords:
Circular colouring, HadwigerNelson problem, Coloring of the plane
Abstract
We consider circular version of the famous NelsonHadwiger 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].