The Heule spindle is the 10-node unit-distance graph illustrated above (Soifer 2024, pp. 696-697) with chromatic number 4. It can be used to construct unit-distance graphs having chromatic number 5 without the use of Moser spindles. It appears as a subgraph in the de Grey graph, Exoo-Ismailescu graphs on 49, 51, and 627 nodes, Heule graphs, Mixon graphs, and Parts graphs on 510, 525, 529, and 553 vertices. It also appears in the 51-braced dodecagon, 45-braced octagon, 75-braced pentagon, and 21-braced square.
Unlike the Moser spindle, it is a matchstick graph. It is illustrated above in a number of embeddings.
It will be implemented in a future version of the Wolfram Language as GraphData["HeuleSpindle"].
