fourcolour map problem

In topology, a longstanding conjecture asserting that no more than four colours are required to shade in any map such that each adjacent region is coloured differently.First posed in 1852 by Francis Guthrie, a British math student, it was solved by Kenneth Appel and Wolfgang Haken using a computerassisted proof in 1976.
* * *
problem in topology, originally posed in the early 1850s and not solved until 1976, that required finding the minimum number of different colours required to colour a map such that no two adjacent regions (i.e., with a common boundary segment) are of the same colour. Three colours are not enough, since one can draw a map of four regions with each region contacting the three other regions. It had been proved mathematically by the English attorney Alfred Bray Kempe in 1879 that five colours will always suffice; and no map had ever been found on which four colours would not do. As is often the case in mathematics, consideration of the problem provided the impetus for the discovery of related results in topology and combinatorics. A similar problem had been solved for the seemingly more complicated situation of a map drawn on a torus (doughnutshaped surface), where seven colours were known to be the minimum.The fourcolour problem was solved in 1977 by a group of mathematicians at the University of Illinois, directed by Kenneth Appel and Wolfgang Haken, after four years of unprecedented synthesis of computer search and theoretical reasoning. Appel and Haken created a catalog of 1,936 “unavoidable” configurations, at least one of which must be present in any graph, no matter how large. Then they showed how each of these configurations could be reduced to a smaller one so that, if the smaller one could be coloured with four colours, so could the original catalog configuration. Thus, if there were a map that could not be coloured with four colours, they could use their catalog to find a smaller map that also could not be fourcoloured, and then a smaller one still, and so on. Eventually this reduction process would lead to a map with only three or four regions that, supposedly, could not be coloured with four colours. This absurd result, which is derived from the hypothesis that a map requiring more than four colours might exist, leads to the conclusion that no such map can exist. All maps are in fact fourcolourable.The strategy involved in this proof dates back to the 1879 paper of Kempe, who produced a short list of unavoidable configurations and then showed how to reduce each to a smaller case. Appel and Haken replaced Kempe's brief list with their catalog of 1,936 cases, each involving up to 500,000 logical options for full analysis. Their complete proof, itself several hundred pages long, required more than 1,000 hours of computer calculations.The fact that the proof of the fourcolour problem had a substantial component that relied on a computer and that could not be verified by hand led to a considerable debate among mathematicians about whether the theorem should be considered “proved” in the usual sense. In 1997 other mathematicians reduced the number of unavoidable configurations to 633 and made some simplifications in the argument, without, however, completely eliminating the computer portion of the proof. There remains some hope for an eventual “computerfree” proof.Robert Osserman* * *
Universalium. 2010.
Look at other dictionaries:
Map — /map/, n. Walter, c1140 1209?, Welsh ecclesiastic, poet, and satirist. Also, Mapes /mayps, may peez/. * * * I Graphic representation, drawn to scale and usually on a flat surface, of features usually geographic, geologic, or geopolitical of an… … Universalium
map — mappable, adj. mapper, n. /map/, n., v., mapped, mapping. n. 1. a representation, usually on a flat surface, as of the features of an area of the earth or a portion of the heavens, showing them in their respective forms, sizes, and relationships… … Universalium
MAP — See modified American plan. * * * I Graphic representation, drawn to scale and usually on a flat surface, of features usually geographic, geologic, or geopolitical of an area of the Earth or of any celestial body. Globes are maps represented on… … Universalium
colour — /kul euhr/, n., adj. v.t., v.i. Chiefly Brit. color. Usage. See or1. * * * I Aspect of any object that may be described in terms of hue, brightness, and saturation. It is associated with the visible wavelengths of electromagnetic radiation, which … Universalium
problem — /prob leuhm/, n. 1. any question or matter involving doubt, uncertainty, or difficulty. 2. a question proposed for solution or discussion. 3. Math. a statement requiring a solution, usually by means of a mathematical operation or geometric… … Universalium
four — /fawr, fohr/, n. 1. a cardinal number, three plus one. 2. a symbol of this number, 4 or IV or IIII. 3. a set of this many persons or things. 4. a playing card, die face, or half of a domino face with four pips. 5. fours, Jazz. alternate four bar… … Universalium
Four color theorem — Example of a four colored map A four colori … Wikipedia
4FarbenProblem — Der Vier Farben Satz (früher auch als Vier Farben Vermutung oder Vier Farben Problem bekannt) ist ein mathematischer Satz und besagt, dass vier Farben immer ausreichen, um eine beliebige Landkarte in der euklidischen Ebene so einzufärben, dass… … Deutsch Wikipedia
VierFarbenProblem — Der Vier Farben Satz (früher auch als Vier Farben Vermutung oder Vier Farben Problem bekannt) ist ein mathematischer Satz und besagt, dass vier Farben immer ausreichen, um eine beliebige Landkarte in der euklidischen Ebene so einzufärben, dass… … Deutsch Wikipedia
Tube map — as of April 2011. Part of a series of articles on The Tube … Wikipedia