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.

DETAILS

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.

PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.