Mixon 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. De Grey he was able to reduce the size of his smallest example (after a correction) to the 1581-vertex de Grey graph (de Grey 2018).

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 graphs the Mixon graphs, illustrated above.

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 2020).

The Mixon graphs are implemented in the Wolfram Language as GraphData["MixonGraph1577"] etc.