Four Color Theorem

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Four Color Theorem
Field: Graph Theory
Image Created By: Brendan John

Four Color Theorem

This image shows a four coloring and graph representation of the United States.

Basic Description

Suppose 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.

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. Two vertices connected by an edge are said to be adjacent. A graph is planar if it can be drawn with no edges overlapping each other, as shown in the top figure of the diagram to the left.

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 two-dimensional territory with a vertex (a point) 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.

A More Mathematical Explanation

Coloring maps on other figures

Graph theory is further used to analyze maps which live on objec [...]

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 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.

Construction of a torus such that 7 regions are in mutual contact.

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