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).
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).
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.
