Moser Spindle

The Moser spindle is a graph in mathematics named for the brothers Leo Moser and William Moser. It has seven vertices and eleven edges, and it can be drawn so that every edge has exactly the same length, making it a unit distance graph. Because it needs four colors in any proper graph coloring, its existence helps show that coloring the plane so that no two points exactly one unit apart share a color requires at least four colors, a question known as the Hadwiger-Nelson problem. The graph is planar, can be built by the Hajos construction, and is sometimes also called the Hajos graph, though that name has been used for other graphs too. Before a five-coloring example turned up in 2018, the Moser spindle gave the best known lower bound on this coloring problem, whose exact answer is still open, with seven colors as the current best known upper bound.