What percent of the plane can be properly 5- and 6-colored?

TL;DR

This paper introduces new tilings that cover over 99.98% and 95.99% of the Euclidean plane with six and five colors respectively, significantly improving previous coverage records.

Contribution

The authors present the first tilings covering over 99.98% and 95.99% of the plane with six and five colors, respectively, advancing the understanding of plane colorability.

Findings

Over 99.985698% of the plane can be 6-colored.

Over 95.99% of the plane can be 5-colored.

Any unit-distance graph with up to 6992 vertices can be properly 6-colored, and with up to 24 vertices can be 5-colored.

Abstract

We present a tiling of more than 99.985698% of the Euclidean plane with six colors, reducing the previous record for uncovered fraction of the plane by about 12.8%. We also present a tiling of more than 95.99% of the plane with five colors. It is thus shown that any unit-distance graph of order at most 6992 and 24 in the plane can be properly 6-colored and 5-colored, respectively.