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 (2026) also constructed a 5-chromatic, triangle-free, unit-distance graph in three dimensions containing 61 vertices via spindling of a 31-vertex graph. These graphs are illustrated above. This 61-vertex graph graph is much smaller than the Voronov-Neopryatnaya-Dergachev graphs, which are two tetahedron-free graphs on 372 and 972 vertices which have unit-distance embeddings with all vertices on a sphere and having chromatic number 5. Exact coordinates for this graph were found by E. Weisstein (Dec. 19, 2025).

The de Grey graphs on 31, 60, 61, and 126 vertices are implemented in the Wolfram Language as GraphData["DeGreyGraph31"], GraphData["DeGreyGraph60"], GraphData["DeGreyGraph61"], and GraphData["DeGreyGraph126"], respectively.