Computing Small Unit-Distance Graphs with Chromatic Number 5
Abstract
We present a new method for reducing the size of graphs with a given property. Our method, which is based on clausal proof minimization, allowed us to compute several 553-vertex unit-distance graphs with chromatic number 5, while the smallest published unit-distance graph with chromatic number 5 has 1581 vertices. The latter graph was constructed by Aubrey de Grey to show that the chromatic number of the plane is at least 5. The lack of a 4-coloring of our graphs is due to a clear pattern enforced on some vertices. Also, our graphs can be mechanically validated in a second, which suggests that the pattern is based on a reasonably short argument.
Keywords
Cite
@article{arxiv.1805.12181,
title = {Computing Small Unit-Distance Graphs with Chromatic Number 5},
author = {Marijn J. H. Heule},
journal= {arXiv preprint arXiv:1805.12181},
year = {2018}
}
Comments
To appear in Geombinatorics XXVIII(1) in July-2018, a special issue dedicated to 5-chromatic unit-distance graphs