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- Four_color_theorem abstract "In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a point that also belongs to Arizona and Colorado, are not.Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May (Wilson 2002, 2), “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property.”Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century (Heawood 1890); however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.The four color theorem was 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. (If they did appear, you could make a smaller counter-example.) Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples existed because any must contain, yet not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand (Swart 1980). Since then the proof has gained wider acceptance, although doubts remain (Wilson 2002, 216–222).To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software.".
- Four_color_theorem thumbnail Four_Colour_Map_Example.svg?width=300.
- Four_color_theorem wikiPageExternalLink Swart697-707.pdf.
- Four_color_theorem wikiPageExternalLink bcc.pdf.
- Four_color_theorem wikiPageExternalLink 4colproof.pdf.
- Four_color_theorem wikiPageExternalLink The_four_colour_theorem.html.
- Four_color_theorem wikiPageExternalLink thomas.pdf.
- Four_color_theorem wikiPageExternalLink tx081101382p.pdf.
- Four_color_theorem wikiPageExternalLink discuss.php?ip=&url=plik&nIdA=21787&sTyp=HTML&nIdSesji=-1.
- Four_color_theorem wikiPageExternalLink 5.
- Four_color_theorem wikiPageID "10949".
- Four_color_theorem wikiPageRevisionID "604507072".
- Four_color_theorem authorlink "Arthur Cayley".
- Four_color_theorem first "Arthur".
- Four_color_theorem hasPhotoCollection Four_color_theorem.
- Four_color_theorem id "p/f040970".
- Four_color_theorem last "Cayley".
- Four_color_theorem title "Blanuša snarks".
- Four_color_theorem title "Four-colour problem".
- Four_color_theorem title "Map coloring".
- Four_color_theorem urlname "BlanusaSnarks".
- Four_color_theorem urlname "MapColoring".
- Four_color_theorem year "1879".
- Four_color_theorem subject Category:Graph_coloring.
- Four_color_theorem subject Category:Planar_graphs.
- Four_color_theorem subject Category:Theorems_in_graph_theory.
- Four_color_theorem type Abstraction100002137.
- Four_color_theorem type Communication100033020.
- Four_color_theorem type Graph107000195.
- Four_color_theorem type Message106598915.
- Four_color_theorem type PlanarGraphs.
- Four_color_theorem type Proposition106750804.
- Four_color_theorem type Statement106722453.
- Four_color_theorem type Theorem106752293.
- Four_color_theorem type TheoremsInDiscreteMathematics.
- Four_color_theorem type VisualCommunication106873252.
- Four_color_theorem comment "In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions.".
- Four_color_theorem label "Four color theorem".
- Four_color_theorem label "Teorema das quatro cores".
- Four_color_theorem label "Teorema de los cuatro colores".
- Four_color_theorem label "Teorema dei quattro colori".
- Four_color_theorem label "Théorème des quatre couleurs".
- Four_color_theorem label "Twierdzenie o czterech barwach".
- Four_color_theorem label "Vier-Farben-Satz".
- Four_color_theorem label "Vierkleurenstelling".
- Four_color_theorem label "Проблема четырёх красок".
- Four_color_theorem label "مبرهنة الألوان الأربعة".
- Four_color_theorem label "四色定理".
- Four_color_theorem label "四色定理".
- Four_color_theorem sameAs Problém_čtyř_barev.
- Four_color_theorem sameAs Vier-Farben-Satz.
- Four_color_theorem sameAs Θεώρημα_των_τεσσάρων_χρωμάτων.
- Four_color_theorem sameAs Teorema_de_los_cuatro_colores.
- Four_color_theorem sameAs Lau_koloreen_teorema.
- Four_color_theorem sameAs Théorème_des_quatre_couleurs.
- Four_color_theorem sameAs Teorema_dei_quattro_colori.
- Four_color_theorem sameAs 四色定理.
- Four_color_theorem sameAs 4색정리.
- Four_color_theorem sameAs Vierkleurenstelling.
- Four_color_theorem sameAs Twierdzenie_o_czterech_barwach.
- Four_color_theorem sameAs Teorema_das_quatro_cores.
- Four_color_theorem sameAs m.02yq7.
- Four_color_theorem sameAs Q184410.
- Four_color_theorem sameAs Q184410.
- Four_color_theorem sameAs Four_color_theorem.
- Four_color_theorem wasDerivedFrom Four_color_theorem?oldid=604507072.
- Four_color_theorem depiction Four_Colour_Map_Example.svg.
- Four_color_theorem isPrimaryTopicOf Four_color_theorem.