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

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

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arxiv 2010.12668 v1 pith:BLZEHZTD submitted 2020-10-23 math.CO math.MG

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classification math.COmath.MG

keywords planecoloredcolorsproperlytilingeuclideanfivefraction

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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.

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Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?
cs.LG 2025-01 conditional novelty 6.0 of 10

A neural network relaxation of geometric coloring constraints produced new plane colorings, including an almost 5-coloring covering all but 3.74% of the plane, improving known bounds for Hadwiger-Nelson variants.

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