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# de Grey Graphs

De Grey (2018) found the first examples of unit-distance graphs with chromatic number 5, thus demonstrating that the solution to the Hadwiger-Nelson problem (i.e., the chromatic number of the plane) is at least 5. While de Grey's original graph contained vertices, he was able to reduce this number (after a correction) to the 1581-vertex graph illustrated above (de Grey 2018), referred to in this work as the de Grey graph.

A few days after the original preprint was published, Mixon (2018) constructed a similar 1585-vertex graph, the removal of 8 vertices from which led to an even smaller 1577-vertex graph. This work terms these two graphs the Mixon graphs.

Smaller unit-distance graphs with chromatic number 5, here called the Heule graphs and Parts graphs, were computationally derived from the de Grey graph by Marijn Heule and Jaan Parts between 2018 and 2020. As of 2022, the smallest of these is the 509-vertex Parts graph (Parts 2020a).

The de Grey graph is implemented in the Wolfram Language as GraphData["DeGreyGraph"].

Additional graphs due to de Grey are 59- and 60-vertex graphs with chromatic number 6 that are unit-distance in 3 dimensions (but not 2) and a 126-vertex graph discussed by Parts (2020b).

de Grey-Haugstrup Graphs, Golomb Graph, Hadwiger-Nelson Problem, Heule Graphs, Mixon Graphs, Moser Spindle, Parts Graphs, Unit-Distance Graph

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## References

de Grey, A. D. N. J. "The Chromatic Number of the Plane Is at Least 5." Geombinatorics 28, No. 1, 18-31, 2018.Heule, M. J. H. "Computing Small Unit-Distance Graphs with Chromatic Number 5." Geombinatorics 28, 32-50, 2018.Lamb, E. "Decades-Old Graph Problem Yields to Amateur Mathematician." Quanta Mag. Apr. 17, 2018. https://www.quantamagazine.org/decades-old-graph-problem-yields-to-amateur-mathematician-20180417/.Mixon, D. G. "The Chromatic Number of the Plane Is at Least 5."Apr. 10, 2018. https://dustingmixon.wordpress.com/2018/04/10/the-chromatic-number-of-the-plane-is-at-least-5/.Mixon, D. G. "Polymath16, First Thread: Simplifying De Grey's Graph." 14 Apr 2018. https://dustingmixon.wordpress.com/2018/04/14/polymath16-first-thread-simplifying-de-greys-graph/.Parts, J. "Graph Minimization, Focusing on the Example of 5-Chromatic Unit-Distance Graphs in the Plane." Geombinatorics 29, No. 4, 137-166, 2020a.Parts, J. "A Small 6-Chromatic Two-Distance Graph in the Plane." 23 Oct 2020b. https://arxiv.org/abs/2010.12656.PolyMath. "Hadwiger-Nelson Problem." http://michaelnielsen.org/polymath1/index.php?title=Hadwiger-Nelson_problem.Soifer, A. "Aubrey D. N. J. de Grey's Breakthrough" and "De Grey's Construction." Ch. 51-52 in The New Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators, 2nd ed. New York: Springer, pp. 657-674, 2024.

## Cite this as:

Weisstein, Eric W. "de Grey Graphs." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/deGreyGraphs.html