Earth–Moon problem
unsolved problem on graph coloring
The Earth–Moon problem is an unsolved problem on graph coloring in mathematics. It is an extension of the planar map coloring problem (solved by the four color theorem), and was posed by Gerhard Ringel in 1959. An intuitive form of the problem asks how many colors are needed to color political maps of the Earth and Moon, in a hypothetical future where each Earth country has a Moon colony which must be given the same color. In mathematical terms, it seeks the chromatic number of biplanar graphs. It is known that this number is at least 9 and at most 12.
The Earth–Moon problem has been extended to analogous problems of coloring maps on any number of planets. For this extension the lower bounds and upper bounds on the number of colors are closer, within two of each other. One real-world application of the Earth–Moon problem involves testing printed circuit boards.
Formulation and history
In the map coloring problem, finitely many simply connected regions in the Euclidean plane or a topologically equivalent space, such as countries on the surface of the Earth, are to be colored so that, when two regions share a boundary of nonzero length, they have different colors. It can be transformed into a graph coloring problem by making a vertex for each region and an edge for each two neighboring regions, producing a planar graph whose vertices are to be colored.
“Earth–Moon problem” enters the record as unsolved problem on graph coloring. Crown Archives preserves that source wording while asking what Earth, Moon and problem can confirm, complicate or overturn.
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This entry incorporates text from “Earth–Moon problem” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.