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The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken in a computer-aided proof. This came after many false ...
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 ...
Robertson, Seymour & Thomas (1993) proved the conjecture for =, also using the four color theorem; their paper with this proof won the 1994 Fulkerson Prize. It follows from their proof that linklessly embeddable graphs, a three-dimensional analogue of planar graphs, have chromatic number at most five. [3]
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.
While Deligne's final paper proving these conjectures were "only" about 30 pages long, it depended on background results in algebraic geometry and étale cohomology that Deligne estimated to be about 2000 pages long. 1974 4-color theorem. Appel and Haken's proof of this took 139 pages, and also depended on long computer calculations.
Francis Guthrie (born 22 January 1831 in London; d. 19 October 1899 in Claremont, Cape Town) was a Cape Colony mathematician and botanist who first posed the Four Colour Problem in 1852. He studied mathematics under Augustus De Morgan, and botany under John Lindley at University College London. Guthrie obtained his B.A. in 1850, and LL.B. in ...
According to Jensen & Toft (1995), the problem was first formulated by Nelson in 1950, and first published by Gardner (1960). Hadwiger (1945) had earlier published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961).
Next follow chapters on spanning trees and Cayley's formula, chemical graph theory and graph enumeration, and planar graphs, Kuratowski's theorem, and Euler's polyhedral formula. There are three chapters on the four color theorem and graph coloring, a chapter on algebraic graph theory, and a final chapter on graph factorization. Appendices ...