Four Color Theorem

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Four Color Theorem

Four coloring and graph representation of the United States.

Basic Description

How many colors are needed to color the territories of a map, if all the territories that share a border must be of different colors?

It turns out that only four colors are needed to color such a two-dimensional map. It has taken over a century for a correct proof of this fact to emerge, and the proof can currently only be carried out with the aid of computers. An example of a map colored with only 4 colors is the map of The United States on this page's main image.

A More Mathematical Explanation

[[Image:Graphexample.JPG|thumb|left|200px|Example of a planar graph (top) and a non-planar graph (bot [...]

Example of a planar graph (top) and a non-planar graph (bottom)
Map coloring is an application of Graph Theory, the study of graphs. A graph is informally a collection of points, known as vertices, connected by lines, known as edges. A graph is planar if no two edges overlap each other, as shown in the diagram to the left. Graphs are useful to analyze map coloring because a map can be easily converted into a planar graph by representing each territory with a vertex and each border with an edge, as in this page's main image.

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