Four Color Theorem
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Revision as of 12:54, 22 June 2009
|Four Color Theorem|
Four Color Theorem
- This image shows a four coloring and graph representation of the United States.
Basic DescriptionSuppose we have a map in which no single territory is made up of disconnected regions. How many colors are needed to color the territories of this map, if all the territories that share a border segment 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 currently known proofs are so long that they can only be checked with the aid of computers. An example of a map colored with only 4 colors is the map of The United States in 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 [...]
Graphs are useful for analyzing map colorings because a map containing only connected territories can easily be converted into a planar graph by representing each territory with a vertex and each border segment with an edge, as in this page's main image. An edge is not drawn between two territories that share only a corner, such as between Utah and New Mexico. The four color theorem states that the vertices of any planar graph can be colored with at most four colors such that no adjacent vertices are the same color. Since maps can be represented by planar graphs, this theorem is equivalent to saying any map with only connected territories can be colored with at most four colors, such that no territories of the same color will share a border segment.
Coloring maps on other figures
Graph theory is further used to analyze maps which live on objects other than a flat two-dimensional surface. For example, a map which exists on the surface of a three-dimensional sphere only needs four colors, by the following argument: project the three dimensional map onto the sphere using a stereographic projection, color the two-dimensional projection using at most four colors, then color the corresponding regions of the original 3-D map with these same colors.
A torus on the other hand requires seven colors, since it is possible to draw a torus with seven regions in mutual contact, as shown in the below diagram.
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