Chromatic properties of the Euclidean plane

TL;DR

This paper investigates the chromatic number of certain geometric graphs in the plane, providing new bounds and restrictions on proper colorings based on distance intervals and tiling-inspired conditions.

Contribution

It introduces new bounds for the chromatic number of graphs with edges in specific distance ranges and explores coloring restrictions when regions meet along lines.

Findings

For small epsilon, the chromatic number of H(epsilon) is between 6 and 7.

Proper coloring with regions meeting along a line requires at least 5 colors.

Improves previous lower bounds on the chromatic number for these geometric graphs.

Abstract

Let $G$ be the unit distance graph in the plane. A well-known problem in combinatorial geometry is that of determining the chromatic number of $G$. It is known that $4\le \chi(G)\le 7$. The upper bound of 7 is obtained using tilings of the plane. The present paper studies two problems where we seek proper colourings of $G$, adding restrictions inspired by tilings: Let $H(\epsilon)$ be the graph whose vertices are the points of ${\mathbb R}^2$, with an edge between two points if their distance lies in the interval $[1,1+\epsilon]$. We show that for small $\epsilon$, $0<\epsilon\le \frac{3\sqrt{2}}{4}-1$, we have $6\le \chi(H(\epsilon))\le 7$. This improves the result of Exoo and Grytczuk et al. that $5\le \chi(H(\epsilon))$ for small $\epsilon$. Suppose that $G$ is properly coloured, but so that two solidly coloured regions meet along a straight line in some neighbourhood. Then at least 5 colours must be used.