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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.
Foster's theorem ; Four color theorem (graph theory) Four functions theorem (combinatorics) Four-vertex theorem (differential geometry) Fourier inversion theorem (harmonic analysis) Fourier theorem (harmonic analysis) Franel–Landau theorem (number theory) Fraňková–Helly selection theorem (mathematical analysis) Fredholm's theorem (linear ...
Kempe's proof did, however, suffice to show the weaker five color theorem. The four-color theorem was eventually proved by Kenneth Appel and Wolfgang Haken in 1976. [2] Schröder–Bernstein theorem. In 1896 Schröder published a proof sketch [3] which, however, was shown to be faulty by Alwin Reinhold Korselt in 1911 [4] (confirmed by ...
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.
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 ...
The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if ...
Haboush's theorem: 1976: Kenneth Appel and Wolfgang Haken: Four color theorem: graph colouring: Traditionally called a "theorem", long before the proof. 1976: Daniel Quillen; and independently by Andrei Suslin: Serre's conjecture on projective modules: polynomial rings: Quillen–Suslin theorem: 1977: Alberto Calderón: Denjoy's conjecture ...
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 ...
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