Five color theorem
The five color theorem is a result from graph theory that given a plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than five colors in such a way that no two adjacent regions receive the same color.
The five color theorem is implied by the stronger four color theorem, but is considerably easier to prove. It was based on a failed attempt at the four color proof by Alfred Kempe in 1879. Percy John Heawood found an error 11 years later, and proved the five color theorem based on Kempe's work. The four color theorem was finally proven by Kenneth Appel and Wolfgang Haken at the University of Illinois, with the aid of a computer. They were assisted in some algorithmic work by John A. Koch.
First of all, one associates a simple planar graph to the given map, namely one puts a vertex in each region of the map, then connects two vertices with an edge if and only if the corresponding regions share a common border. The problem is then translated into a graph coloring problem: one has to paint the vertices of the graph so that no edge has endpoints of the same color.
Because is a simple planar, i.e. it may be embedded in the plane without intersecting edges, and it does not have two vertices sharing more than one edge, and it doesn't have loops, then it can be shown (using the Euler characteristic of the plane) that it must have a vertex shared by at most five edges. (Note: This is the only place where the five-color condition is used in the proof. If this technique is used to prove the four-color theorem, it will fail on this step. In fact, an icosahedral graph is 5-regular and planar, and thus does not have a vertex shared by at most four edges.) Find such a vertex, and call it .
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