Tiling the plane with hexagons: improved separations for $k$-colourings

TL;DR

This paper improves the known bounds for the minimum distance between same-colored tiles in infinite hexagonal tilings with more than seven colors, advancing understanding of optimal colorings.

Contribution

It presents new tilings with larger separation distances for many color counts, surpassing previous known bounds.

Findings

New tilings with larger $d(k)$ values for many $k$

Enhanced bounds for $k$-colorings of the honeycomb

Improved understanding of coloring separations in hexagonal tilings

Abstract

It has been common knowledge since 1950 that seven colours can be assigned to tiles of an infinite honeycomb with cells of unit diameter such that no two tiles of the same colour are closer than $d(7)=\frac{\sqrt{7}}{2}$ apart. Various authors have described tilings using $k>7$ colours, giving corresponding values for $d(k)$, but it is generally unknown whether these are the largest possible for a given $k$. Here, for many $k$, we describe tilings with larger values of $d(k)$ than previously reported.