Coloring distance graphs on the plane

TL;DR

This paper investigates the minimal number of colors needed to color the Euclidean plane so that points at certain distances have different colors, improving bounds and providing new colorings for specific distance ranges.

Contribution

It offers improved bounds on the chromatic number for distance graphs on the plane and introduces the first 8-coloring for larger distance intervals.

Findings

Determined minimal colors for two specific ranges of distance intervals.

Provided the first 8-coloring for larger distance values.

Established bounds and exact values for coloring bounded regions like annuli.

Abstract

We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of the plane in such a way that pairs of points at distance in the interval $[1,b]$ get different colors. The classic Hadwiger-Nelson problem is a special case of this question -- obtained by taking $b=1$. The main results of the paper are improved lower and upper bounds on the number of colors for some values of $b$. In particular, we determine the minimal number of colors for two ranges of values of $b$ - one of which is enlarging an interval presented by Exoo and the second is completely new. Up to our knowledge, these are the only known families of distance graphs on the plane with a determined nontrivial chromatic number. Moreover, we present the first $8$-coloring for $b$ larger than values of $b$ for the known $7$-colorings. As a byproduct, we give some bounds and exact values for bounded parts of the plane, specifically by coloring certain annuli.