11520

Moser Spindles, Golomb Graphs and Root 33

If the plane is divided into colors, what is the least such that any two points a unit distance apart have different colors? This is the unsolved Hadwiger–Nelson problem, whose answer is 4, 5, 6 or 7 colors.
A -coloring of a graph colors the vertices of with colors so that no two vertices of the same color are connected by an edge. The chromatic number of is the least such .
In other words, the Hadwiger–Nelson problem asks for the chromatic number of the infinite graph whose vertices are all the points in the plane and whose edges are all possible unit segments.
The Moser spindle and the Golomb graph are finite unit distance graphs with chromatic number 4. Whether there is a finite unit distance graph with chromatic number 5 is an unsolved problem.
Vertices of both the Moser spindle and Golomb graph have coordinates in the algebraic field , where is the field of rational numbers. For example, the coordinates of the rightmost vertex of the first Moser spindle are .
This Demonstration shows a unit distance graph in with 2507 vertices, 12502 unit edges, 598 Moser spindle subgraphs and 20 Golomb graph subgraphs. The chromatic number of the graph is at least 4. It might have chromatic number 5, but don't get your hopes up.

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Not all unit distance 4-chromatic graphs are in . For example, Hochberg–O'Donnell's fish graph has vertices that rely on polynomials of degree 12. An exact solution can be found at the end of the Initialization section.
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