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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.
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 ...
A graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case k = 4 of Hadwiger's conjecture In graph theory , the Hadwiger conjecture states that if G {\displaystyle G} is loopless and has no K t {\displaystyle K_{t}} minor then its chromatic number satisfies ...
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.
File:Four Colour Map Example.svg. ... Diagram showing a map coloured with four colours. ... Four color theorem: 04:11, 16 February 2007:
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 ...
The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle ...