A probabilistic Hadwiger-Nelson problem

Jean-Marc Robert; Marc Heinrich; Paul Melotti; Thomas Bourgeat

A probabilistic Hadwiger-Nelson problem

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arxiv 1501.02441 v1 pith:GDCOH3LO submitted 2015-01-11 math.CO

Thomas Bourgeat , Marc Heinrich , Paul Melotti , Jean-Marc Robert This is my paper · ORCID

classification math.CO

keywords probabilitycolorsproblemsomeboundscertaincolordeduced

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If you color a table using k colors, and throw a needle randomly on it, for some proper definition, you get a certain probability that the endpoints will fall on different colors. How can one make this probability maximal? This problem is related to finite graphs having unit-length edges, and some bounds on the optimal probability are deduced.

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Lattice approach to plane colorings
cond-mat.stat-mech 2019-08 conditional novelty 6.0 of 10

Simulated annealing on a unit-range Potts model produces optimal two-to-seven colorings; only the seven-color case converges to zero same-color unit-distance pairs, suggesting the plane's chromatic number is seven.

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