Four Color Theorem
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|ImageDescElem= 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 must be of different colors? | |ImageDescElem= 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 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 can 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 | + | 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 can 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 in this page's main image. |
|ImageDesc=[[Image:Graphexample.JPG|thumb|left|200px|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 no two edges overlap each other, as shown in the diagram to the left. | |ImageDesc=[[Image:Graphexample.JPG|thumb|left|200px|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 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 easily be 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. An edge is not drawn between two territories which share only a corner, such 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 every map can be represented by a planar graph, this theorem is equivalent to saying any map can be colored with at most four colors, | + | Graphs are useful to analyze map coloring because a map can easily be 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. An edge is not drawn between two territories which share only a corner, such 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 every map can be represented by a planar graph, this theorem is equivalent to saying any map can be colored with at most four colors, such that no territories of the same color will share a border. |
|AuthorName=Brendan John | |AuthorName=Brendan John | ||
Revision as of 11:28, 4 June 2009
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Four Color Theorem
- 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 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 can 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 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 to analyze map coloring because a map can easily be 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. An edge is not drawn between two territories which share only a corner, such 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 every map can be represented by a planar graph, this theorem is equivalent to saying any map can be colored with at most four colors, such that no territories of the same color will share a border.
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