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Example of a four-colored map A four-colored map of the states of the United States (ignoring lakes and oceans). In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color.
Kenneth Ira Appel (October 8, 1932 – April 19, 2013) was an American mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana–Champaign, solved one of the most famous problems in mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in ...
English: Diagram showing a map coloured with four colours. See the Four Colour Theorem. ... Category:Four color theorem: 04:11, 16 February 2007: 300 × 400 (14 KB)
Typically, the set S has four elements (the four colours of the four colour theorem), and c is a proper colouring, that is, each pair of adjacent vertices in V are assigned distinct colours. With these additional conditions, a and b are two out of the four colours available, and every element of the ( a , b )-Kempe chain has neighbours in the ...
In 1904, Wernicke introduced the discharging method to prove the following theorem, which was part of an attempt to prove the four color theorem. Theorem: If a planar graph has minimum degree 5, then it either has an edge with endpoints both of degree 5 or one with endpoints of degrees 5 and 6.
A map of the United States using colors to show political divisions using the four color theorem. The first results about graph coloring deal almost exclusively with planar graphs in the form of map coloring .
English: This is a map of France where no two neighboring first-level administrative divisions have the same color, following the guidelines of the four-color theorem, which states that any loopless planar graph can be colored with four or fewer colors. This was created with Mathematica 13.2's function FindVertexColoring and GeoRegionValuePlot ...
An entirely different approach was needed for the much older problem of finding the number of colors needed for the plane or sphere, solved in 1976 as the four color theorem by Haken and Appel. On the sphere the lower bound is easy, whereas for higher genera the upper bound is easy and was proved in Heawood's original short paper that contained ...