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| [[Image:Four Colour Map Example.svg|thumb|Example of a four-colored map]]
| | If you have the desire to procedure settings instantly, loading files instantly, yet the body is logy and torpid, what would we do? If you are a giant "switchboard" that is lack of efficient management program plus efficient housekeeper, what would we do? If you have send your exact commands to the notice, however, the body could not work correctly, what would you do? Yes! We want a full-featured repair registry!<br><br>We have to know some simple and cheap ways which could resolve the problem of your computer plus speed it up. The earlier we fix it, the less damage your computer gets. I usually tell about certain worthwhile techniques that will assist we to accelerate you computer.<br><br>Naturally, the next logical step is to receive these false entries cleaned out. Fortunately, this might be not a difficult task. It is the 2nd thing you need to do when you observed the computer has lost speed. The first will be to make certain there are no viruses or severe spyware present.<br><br>How to fix this issue is to first reinstall the program(s) causing the mistakes. There are a great deal of different programs that use this file, but one might have placed their own faulty version of the file onto the program. By reinstalling any programs which are causing the error, you'll not merely enable the PC to run the program correctly, nevertheless a modern file will be placed onto a program - exiting the computer running because smoothly because possible again. If you try this, plus find it does not work, then you need to look to update the system & any software you have on your PC. This will likely update the Msvcr71.dll file, permitting your computer to read it correctly again.<br><br>After that, I additionally purchased the Regtool [http://bestregistrycleanerfix.com/regzooka regzooka] Software, and it further secure my laptop having program crashes. All my registry issues are fixed, plus I may function peacefully.<br><br>Turn It Off: Chances are should you are like me; then you spend a great deal of time on a computer on a daily basis. Try providing your computer some time to do absolutely nothing; this can sound funny nevertheless should you have an older computer you are asking it to do too much.<br><br>The 'registry' is merely the central database which shops all your settings and options. It's a actually important piece of the XP program, that means that Windows is frequently adding and updating the files inside it. The difficulties occur when Windows actually corrupts & loses a few of these files. This makes a computer run slow, because it attempts difficult to obtain them again.<br><br>By changing the way we use the web you can have access more of your valuable bandwidth. This will eventually give we a faster surfing experience. Here is a link to 3 methods to customize the PC speed online. |
| [[File:Map of United States vivid colors shown.png|thumb|A four-coloring of a map of the states of the United States (ignoring lakes).]]
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| In [[mathematics]], the '''four color theorem''', or the '''four color map theorem''', states that, given any separation of a plane into [[Contiguity#Geography|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.<ref>{{cite journal |title=Formal Proof---The Four-Color Theorem |author=Georges Gonthier |journal=Notices of the AMS |volume=55 |issue=11 |date=December 2008 |pages=1382–1393}}From this paper: Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. A point is a corner of a map if and only if it belongs to the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors.</ref> 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 [[Four Corners Monument|point]] that also belongs to Arizona and Colorado, are not.
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| Despite the motivation from [[map coloring|coloring political maps of countries]], the theorem is not of particular interest to mapmakers. According to an article by the math historian [[Kenneth May]] {{Harv|Wilson|2002|loc=2}}, “Maps utilizing only four colours 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.”
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| 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 {{Harv|Heawood|1890}}; however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false [[counterexample]]s have appeared since the first statement of the four color theorem in 1852.
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| The four color theorem was proven in 1976 by [[Kenneth Appel]] and [[Wolfgang Haken]]. It was the first major [[theorem]] to be [[Computer-assisted proof#List of theorems proved with the help of computer programs|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. Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map (regardless of whether it is a counterexample or not) 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 {{Harv|Swart|1980}}. Since then the proof has gained wider acceptance, although doubts remain {{Harv|Wilson|2002|loc=216–222}}.
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| 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 [[Proof assistant|theorem proving software]].
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| ==Precise formulation of the theorem==
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| The intuitive statement of the four color theorem, i.e. 'that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color', needs to be interpreted appropriately to be correct. First, all corners, points that belong to (technically, are in the closure of) three or more countries, must be ignored. In addition, bizarre maps (using regions of finite area but infinite perimeter) can require more than four colors.<ref>{{cite journal |title=Four Colors Do Not Suffice
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| |author=Hud Hudson
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| |journal=The American Mathematical Monthly
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| |volume=110
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| |number=5
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| |date=May 2003
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| |pages=417–423
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| |jstor=3647828 }}</ref>
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| [[Image:Azerbaijan-CIA WFB Map.png|thumb|Example of a map of [[Azerbaijan]] with non-contiguous regions]]
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| Second, for the purpose of the theorem every "country" has to be a [[Simply connected space|simply connected region]], or [[Contiguity#Geography|contiguous]].
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| In the real world, [[Enclave and exclave|this is not true]] (e.g., Alaska as part of the United States, [[Nakhchivan Autonomous Republic|Nakhchivan]] as part of [[Azerbaijan]], and [[Kaliningrad Oblast|Kaliningrad]] as part of Russia are not contiguous). Because the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map:
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| [[Image:4CT Inadequacy Example.svg|none|center]]
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| In this map, the two regions labeled ''A'' belong to the same country, and must be the same color. This map then requires five colors, since the two ''A'' regions together are contiguous with four other regions, each of which is contiguous with all the others. If ''A'' consisted of three regions, six or more colors might be required; one can construct maps that require an arbitrarily high number of colors. A similar construction also applies if a single color is used for all bodies of water, as is usual on real maps.
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| An easier to state version of the theorem uses [[graph theory]]. The set of regions of a map can be represented more abstractly as an [[undirected graph]] that has a [[vertex (graph theory)|vertex]] for each region and an [[edge (graph theory)|edge]] for every pair of regions that share a boundary segment. This graph is [[planar graph|planar]] (it is important to note that we are talking about the graphs that have some limitations according to the map they are transformed from only): it can be drawn in the plane without crossings by placing each vertex at an arbitrarily chosen location within the region to which it corresponds, and by drawing the edges as curves that lead without crossing within each region from the vertex location to each shared boundary point of the region. Conversely any planar graph can be formed from a map in this way. In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be [[Graph coloring|colored]] with at most four colors so that no two adjacent vertices receive the same color, or for short, "every planar graph is four-colorable" ({{Harvnb|Thomas|1998|p=849}}; {{Harvnb|Wilson|2002}}).
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| [[Image:Four Colour Planar Graph.svg|center]]
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| ==History==<!-- This section is linked from [[Mathematics]] -->
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| ===Early proof attempts===
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| [[August Ferdinand Möbius|Möbius]] mentioned the problem in his lectures as early as 1840.<ref name="rouse_ball_1960">[[W. W. Rouse Ball]] (1960) ''The Four Color Theorem'', in Mathematical Recreations and Essays, Macmillan, New York, pp 222-232.</ref> The conjecture was first proposed on October 23, 1852 <ref name=MacKenzie>Donald MacKenzie, ''Mechanizing Proof: Computing, Risk, and Trust'' (MIT Press, 2004) p103</ref> when [[Francis Guthrie]], while trying to color the map of counties of England, noticed that only four different colors were needed. At the time, Guthrie's brother, Frederick, was a student of [[Augustus De Morgan]] (the former advisor of Francis) at [[University College London]]. Francis inquired with Frederick regarding it, who then took it to De Morgan (Francis Guthrie graduated later in 1852, and later became a professor of mathematics in South Africa). According to De Morgan:
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| <blockquote>"A student of mine [Guthrie] asked me to day to give him a reason for a fact which I did not know was a fact — and do not yet. He says that if a figure be any how divided and the compartments differently coloured so that figures with any portion of common boundary ''line'' are differently coloured — four colours may be wanted but not more — the following is his case in which four colours ''are'' wanted. Query cannot a necessity for five or more be invented… " {{Harv|Wilson|2002|p=18}}</blockquote>
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| "F.G.", perhaps one of the two Guthries, published the question in ''[[Athenaeum (magazine)|The Athenaeum]]'' in 1854,<ref>{{citation|url=http://books.google.com/books?id=Mm1IAAAAYAAJ&pg=PA726|journal=[[Athenaeum (magazine)|The Athenaeum]]|date=June 10, 1854|page=726|author=F. G.|title=Tinting Maps}}.</ref><ref>{{cite arXiv|author = Brendan D. McKay|title = A note on the history of the four-colour conjecture | eprint = 1201.2852 | year = 2012}}</ref> and De Morgan posed the question again in the same magazine in 1860.<ref name=Athenaeum1860>{{citation|journal=[[Athenaeum (magazine)|The Athenaeum]]|date=April 14, 1860|first=Augustus|last=De Morgan (anonymous)|authorlink=Augustus De Morgan|pages=501–503|title=The Philosophy of Discovery, Chapters Historical and Critical. By W. Whewell.}}</ref> Another early published reference by {{harvs|first=Arthur|last=Cayley|authorlink=Arthur Cayley|year=1879|txt}} in turn credits the conjecture to De Morgan.
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| There were several early failed attempts at proving the theorem. De Morgan believed that it followed from a simple fact about four regions, though he didn't believe that fact could be derived from more elementary facts.
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| <blockquote>This arises in the following way. We never need four colours in a neighbourhood unless there be four counties, each of which has boundary lines in common with each of the other three. Such a thing cannot happen with four areas unless one or more of them be inclosed by the rest; and the colour used for the inclosed county is thus set free to go on with. Now this principle, that four areas cannot each have common boundary with all the other three without inclosure, is not, we fully believe, capable of demonstration upon anything more evident and more elementary; it must stand as a postulate.<ref name=Athenaeum1860/></blockquote>
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| One alleged proof was given by [[Alfred Kempe]] in 1879, which was widely acclaimed;<ref name="rouse_ball_1960"/> another was given by [[Peter Guthrie Tait]] in 1880. It was not until 1890 that Kempe's proof was shown incorrect by [[Percy Heawood]], and in 1891 Tait's proof was shown incorrect by [[Julius Petersen]]—each false proof stood unchallenged for 11 years {{Harv|Thomas|1998|p=848}}.
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| In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the [[five color theorem]] {{Harv|Heawood|1890}} and generalized the four color conjecture to surfaces of arbitrary genus—see below.
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| Tait, in 1880, showed that the four color theorem is equivalent to the statement that a certain type of graph (called a [[snark (graph theory)|snark]] in modern terminology) must be non-[[Planar graph|planar]].<ref>{{Citation|authorlink=Peter Guthrie Tait|first=P. G.|last= Tait|title=Remarks on the colourings of maps|journal=Proc. R. Soc. Edinburgh|volume=10|year=1880|pages=729}}</ref>
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| In 1943, [[Hugo Hadwiger]] formulated the [[Hadwiger conjecture (graph theory)|Hadwiger conjecture]] {{Harv|Hadwiger|1943}}, a far-reaching generalization of the four-color problem that still remains unsolved.
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| ===Proof by computer===
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| During the 1960s and 1970s German mathematician [[Heinrich Heesch]] developed methods of using computers to search for a proof. Notably he was the first to use [[Discharging method (discrete mathematics)|discharging]] for proving the theorem, which turned out to be important in the unavoidability portion of the subsequent Appel-Haken proof. He also expanded on the concept of reducibility and, along with Ken Durre, developed a computer test for it. Unfortunately, at this critical juncture, he was unable to procure the necessary supercomputer time to continue his work {{Harv|Wilson|2002}}.
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| Others took up his methods and his computer-assisted approach. While other teams of mathematicians were racing to complete proofs, [[Kenneth Appel]] and [[Wolfgang Haken]] at the [[University of Illinois at Urbana-Champaign|University of Illinois]] announced, on June 21, 1976,<ref>Gary Chartrand and Linda Lesniak, ''Graphs & Digraphs'' (CRC Press, 2005) p221</ref> that they had proven the theorem. They were assisted in some algorithmic work by [[John A. Koch]] {{Harv|Wilson|2002}}.
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| If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors. The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts ({{Harvnb|Wilson|2002}}; {{Harvnb|Appel|Haken|1989}}; {{Harvnb|Thomas|1998|pp=852–853}}):
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| # An ''unavoidable set'' is a set of configurations such that every map must have at least one configuration from this set.
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| # A ''reducible configuration'' is an arrangement of countries that cannot occur in a minimal counterexample. If a map contains a reducible configuration, then the map can be reduced to a smaller map. This smaller map has the condition that if it can be colored with four colors, then the original map can also. This implies that if the original map cannot be colored with four colors the smaller map can't either and so the original map is not minimal.
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| Using mathematical rules and procedures based on properties of reducible configurations, Appel and Haken found an unavoidable set of reducible configurations, thus proving that a minimal counterexample to the four-color conjecture could not exist. Their proof reduced the infinitude of possible maps to 1,936 reducible configurations (later reduced to 1,476) which had to be checked one by one by computer and took over a thousand hours. This reducibility part of the work was independently double checked with different programs and computers. However, the unavoidability part of the proof was verified in over 400 pages of [[microfiche]], which had to be checked by hand {{Harv|Appel|Haken|1989}}.
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| Appel and Haken's announcement was widely reported by the news media around the world, and the math department at the [[University of Illinois]] used a postmark stating "Four colors suffice." At the same time the unusual nature of the proof—it was the first major theorem to be proven with extensive computer assistance—and the complexity of the human-verifiable portion, aroused considerable controversy {{Harv|Wilson|2002}}.
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| In the early 1980s, rumors spread of a flaw in the Appel-Haken proof. [[Ulrich Schmidt]] at [[RWTH Aachen]] examined Appel and Haken's proof for his master's thesis {{Harv|Wilson|2002|loc=225}}. He had checked about 40% of the unavoidability portion and found a significant error in the discharging procedure {{Harv|Appel|Haken|1989}}. In 1986, Appel and Haken were asked by the editor of ''[[Mathematical Intelligencer]]'' to write an article addressing the rumors of flaws in their proof. They responded that the rumors were due to a "misinterpretation of [Schmidt's] results" and obliged with a detailed article {{Harv|Wilson|2002|loc=225–226}}. Their [[magnum opus]], ''Every Planar Map is Four-Colorable'', a book claiming a complete and detailed proof (with a microfiche supplement of over 400 pages), appeared in 1989 and explained Schmidt's discovery and several further errors found by others {{Harv|Appel|Haken|1989}}.
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| ===Simplification and verification===
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| Since the proving of the theorem, efficient algorithms have been found for 4-coloring maps requiring only [[Big O notation|O]](''n''<sup>2</sup>) time, where ''n'' is the number of vertices. In 1996, [[Neil Robertson (mathematician)|Neil Robertson]], [[Daniel P. Sanders]], [[Paul Seymour (mathematician)|Paul Seymour]], and [[Robin Thomas (mathematician)|Robin Thomas]] created a [[Polynomial time|quadratic time]] algorithm, improving on a [[Quartic function|quartic]] algorithm based on Appel and Haken’s proof ({{Harvnb|Thomas|1995}}; {{Harvnb|Robertson|Sanders|Seymour|Thomas|1996}}). This new proof is similar to Appel and Haken's but more efficient because it reduces the complexity of the problem and requires checking only 633 reducible configurations. Both the unavoidability and reducibility parts of this new proof must be executed by computer and are impractical to check by hand {{Harv|Thomas|1998|pp=852–853}}. In 2001, the same authors announced an alternative proof, by proving the [[snark theorem]] ({{Harvnb|Thomas}}; {{Harvnb|Pegg|Melendez|Berenguer|Sendra|2002}}).
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| In 2005, [[Benjamin Werner]] and [[Georges Gonthier]] formalized a proof of the theorem inside the [[Coq]] proof assistant. This removed the need to trust the various computer programs used to verify particular cases; it is only necessary to trust the Coq kernel {{Harv|Gonthier|2008}}.
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| ==Summary of proof ideas==
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| The following discussion is a summary based on the introduction to Appel and Haken's book ''Every Planar Map is Four Colorable'' {{Harv|Appel|Haken|1989}}. Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it. The explanation here is reworded in terms of the modern graph theory formulation above.
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| Kempe's argument goes as follows. First, if planar regions separated by the graph are not ''triangulated'', i.e. do not have exactly three edges in their boundaries, we can add edges without introducing new vertices in order to make every region triangular, including the unbounded outer region. If this [[Planar graph|triangulated graph]] is colorable using four colors or fewer, so is the original graph since the same coloring is valid if edges are removed. So it suffices to prove the four color theorem for triangulated graphs to prove it for all planar graphs, and without loss of generality we assume the graph is triangulated.
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| Suppose ''v'', ''e'', and ''f'' are the number of vertices, edges, and regions (faces). Since each region is triangular and each edge is shared by two regions, we have that 2''e'' = 3''f''. This together with [[Euler characteristic#Planar graphs|Euler's formula]] ''v'' − ''e'' + ''f'' = 2 can be used to show that 6''v'' − 2''e'' = 12. Now, the ''degree'' of a vertex is the number of edges abutting it. If ''v''<sub>''n''</sub> is the number of vertices of degree ''n'' and ''D'' is the maximum degree of any vertex,
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| :<math>6v - 2e = 6\sum_{i=1}^D v_i - \sum_{i=1}^D iv_i = \sum_{i=1}^D (6 - i)v_i = 12.</math> | |
| But since 12 > 0 and 6 − ''i'' ≤ 0 for all ''i'' ≥ 6, this demonstrates that there is at least one vertex of degree 5 or less.
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| If there is a graph requiring 5 colors, then there is a ''minimal'' such graph, where removing any vertex makes it four-colorable. Call this graph ''G''. ''G'' cannot have a vertex of degree 3 or less, because if ''d''(''v'') ≤ 3, we can remove ''v'' from ''G'', four-color the smaller graph, then add back ''v'' and extend the four-coloring to it by choosing a color different from its neighbors.
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| Kempe also showed correctly that ''G'' can have no vertex of degree 4. As before we remove the vertex ''v'' and four-color the remaining vertices. If all four neighbors of ''v'' are different colors, say red, green, blue, and yellow in clockwise order, we look for an alternating path of vertices colored red and blue joining the red and blue neighbors. Such a path is called a [[Kempe chain]]. There may be a Kempe chain joining the red and blue neighbors, and there may be a Kempe chain joining the green and yellow neighbors, but not both, since these two paths would necessarily intersect, and the vertex where they intersect cannot be colored. Suppose it is the red and blue neighbors that are not chained together. Explore all vertices attached to the red neighbor by red-blue alternating paths, and then reverse the colors red and blue on all these vertices. The result is still a valid four-coloring, and ''v'' can now be added back and colored red.
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| This leaves only the case where ''G'' has a vertex of degree 5; but Kempe's argument was flawed for this case. Heawood noticed Kempe's mistake and also observed that if one was satisfied with proving only five colors are needed, one could run through the above argument (changing only that the minimal counterexample requires 6 colors) and use Kempe chains in the degree 5 situation to prove the [[five color theorem]].
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| In any case, to deal with this degree 5 vertex case requires a more complicated notion than removing a vertex. Rather the form of the argument is generalized to considering ''configurations'', which are connected subgraphs of ''G'' with the degree of each vertex (in G) specified. For example, the case described in degree 4 vertex situation is the configuration consisting of a single vertex labelled as having degree 4 in ''G''. As above, it suffices to demonstrate that if the configuration is removed and the remaining graph four-colored, then the coloring can be modified in such a way that when the configuration is re-added, the four-coloring can be extended to it as well. A configuration for which this is possible is called a ''reducible configuration''. If at least one of a set of configurations must occur somewhere in G, that set is called ''unavoidable''. The argument above began by giving an unavoidable set of five configurations (a single vertex with degree 1, a single vertex with degree 2, ..., a single vertex with degree 5) and then proceeded to show that the first 4 are reducible; to exhibit an unavoidable set of configurations where every configuration in the set is reducible would prove the theorem.
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| Because ''G'' is triangular, the degree of each vertex in a configuration is known, and all edges internal to the configuration are known, the number of vertices in ''G'' adjacent to a given configuration is fixed, and they are joined in a cycle. These vertices form the ''ring'' of the configuration; a configuration with ''k'' vertices in its ring is a ''k''-ring configuration, and the configuration together with its ring is called the ''ringed configuration''. As in the simple cases above, one may enumerate all distinct four-colorings of the ring; any coloring that can be extended without modification to a coloring of the configuration is called ''initially good''. For example, the single-vertex configuration above with 3 or less neighbors were initially good. In general, the surrounding graph must be systematically recolored to turn the ring's coloring into a good one, as was done in the case above where there were 4 neighbors; for a general configuration with a larger ring, this requires more complex techniques. Because of the large number of distinct four-colorings of the ring, this is the primary step requiring computer assistance.
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| Finally, it remains to identify an unavoidable set of configurations amenable to reduction by this procedure. The primary method used to discover such a set is the [[Discharging method (discrete mathematics)|method of discharging]]. The intuitive idea underlying discharging is to consider the planar graph as an electrical network. Initially positive and negative "electrical charge" is distributed amongst the vertices so that the total is positive.
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| Recall the formula above:
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| :<math>\sum_{i=1}^D (6 - i)v_i = 12.</math>
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| Each vertex is assigned an initial charge of 6-deg(''v''). Then one "flows" the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the ''discharging procedure''. Since charge is preserved, some vertices still have positive charge. The rules restrict the possibilities for configurations of positively-charged vertices, so enumerating all such possible configurations gives an unavoidable set.
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| As long as some member of the unavoidable set is not reducible, the discharging procedure is modified to eliminate it (while introducing other configurations). Appel and Haken's final discharging procedure was extremely complex and, together with a description of the resulting unavoidable configuration set, filled a 400-page volume, but the configurations it generated could be checked mechanically to be reducible. Verifying the volume describing the unavoidable configuration set itself was done by peer review over a period of several years.
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| A technical detail not discussed here but required to complete the proof is ''[[Immersion (mathematics)|immersion]] reducibility''.
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| ==False disproofs==
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| The four color theorem has been notorious for attracting a large number of false proofs and disproofs in its long history. At first, ''[[The New York Times]]'' refused as a matter of policy to report on the Appel–Haken proof, fearing that the proof would be shown false like the ones before it {{Harv|Wilson|2002}}. Some alleged proofs, like Kempe's and Tait's mentioned above, stood under public scrutiny for over a decade before they were exposed. But many more, authored by amateurs, were never published at all.
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| {{double image|right|4CT_Non-Counterexample 1.svg|150|4CT_Non-Counterexample 2.svg|150|The map (left) has been colored with five colors, and it is necessary to change at least four of the ten regions to obtain a coloring with only four colors (right).}}
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| Generally, the simplest, though invalid, counterexamples attempt to create one region which touches all other regions. This forces the remaining regions to be colored with only three colors. Because the four color theorem is true, this is always possible; however, because the person drawing the map is focused on the one large region, they fail to notice that the remaining regions can in fact be colored with three colors.
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| This trick can be generalized: there are many maps where if the colors of some regions are selected beforehand, it becomes impossible to color the remaining regions without exceeding four colors. A casual verifier of the counterexample may not think to change the colors of these regions, so that the counterexample will appear as though it is valid.
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| Perhaps one effect underlying this common misconception is the fact that the color restriction is not [[Transitive relation|transitive]]: a region only has to be colored differently from regions it touches directly, not regions touching regions that it touches. If this were the restriction, planar graphs would require arbitrarily large numbers of colors.
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| Other false disproofs violate the assumptions of the theorem in unexpected ways, such as using a region that consists of multiple disconnected parts, or disallowing regions of the same color from touching at a point.
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| ==Generalizations==
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| [[Image:Torus with seven colours.svg|thumb|300px|By joining the single arrows together and the double arrows together, one obtains a [[torus]] with seven mutually touching regions; therefore seven colors are necessary]]
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| [[Image:Projection color torus.png|480px|thumb|This construction shows the torus divided into the maximum of seven regions, every one of which touches every other.]]
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| The four-color theorem applies not only to finite planar graphs, but also to [[infinite graph]]s that can be drawn without crossings in the plane, and even more generally to infinite graphs (possibly with an uncountable number of vertices) for which every finite subgraph is planar. To prove this, one can combine a proof of the theorem for finite planar graphs with the [[De Bruijn–Erdős theorem (graph theory)|De Bruijn–Erdős theorem]] stating that, if every finite subgraph of an infinite graph is ''k''-colorable, then the whole graph is also ''k''-colorable {{harvtxt|Nash-Williams|1967}}. This can also be seen as an immediate consequence of [[Kurt Gödel]]'s [[compactness theorem]] for [[First-Order Logic]], simply by expressing the colorability of an infinite graph with a set of logical formulae.
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| One can also consider the coloring problem on surfaces other than the plane ([[#Reference-Mathworld-Map coloring|Weisstein]]). The problem on the sphere or cylinder is equivalent to that on the plane. For closed (orientable or non-orientable) surfaces with positive [[Genus (mathematics)|genus]], the maximum number ''p'' of colors needed depends on the surface's [[Euler characteristic]] χ according to the formula
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| :<math>p=\left\lfloor\frac{7 + \sqrt{49 - 24 \chi}}{2}\right\rfloor,</math>
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| where the outermost brackets denote the [[floor function]].
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| Alternatively, for an [[orientable]] surface the formula can be given in terms of the genus of a surface, ''g'':
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| ::<math>p=\left\lfloor\frac{7 + \sqrt{1 + 48g }}{2}\right\rfloor</math> ([[#Reference-Mathworld-Map coloring|Weisstein]]).
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| This formula, the [[Heawood conjecture]], was conjectured by [[P.J. Heawood]] in 1890 and proven by [[Gerhard Ringel]] and [[J. T. W. Youngs]] in 1968. The only exception to the formula is the [[Klein bottle]], which has Euler characteristic 0 (hence the formula gives p = 7) and requires 6 colors, as shown by P. Franklin in 1934 ([[#Reference-Mathworld-Map coloring|Weisstein]]). | |
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| For example, the [[torus]] has Euler characteristic χ = 0 (and genus ''g'' = 1) and thus ''p'' = 7, so no more than 7 colors are required to color any map on a torus. The [[Szilassi polyhedron]] is an example that requires seven colors.
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| A [[Möbius strip]] requires six colors ([[#Reference-Mathworld-Map coloring|Weisstein]]) as do [[1-planar graph]]s (graphs drawn with at most one simple crossing per edge) {{harv|Borodin|1984}}. If both the vertices and the faces of a planar graph are colored, in such a way that no two adjacent vertices, faces, or vertex-face pair have the same color, then again at most six colors are needed {{harv|Borodin|1984}}.
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| There is no obvious extension of the coloring result to three-dimensional solid regions. By using a set of ''n'' flexible rods, one can arrange that every rod touches every other rod. The set would then require ''n'' colors, or ''n''+1 if you consider the empty space that also touches every rod. The number ''n'' can be taken to be any integer, as large as desired. Such examples were known to Fredrick Guthrie in 1880 {{Harv|Wilson|2002}}. Even for axis-parallel [[cuboid]]s (considered to be adjacent when two cuboids share a two-dimensional boundary area) an unbounded number of colors may be necessary ({{harvnb|Reed|Allwright|2008}}; {{harvtxt|Magnant|Martin|2011}}).
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| ==See also==
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| ;[[Graph coloring]]
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| :the problem of finding optimal colorings of graphs that are not necessarily planar.
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| ;[[Grötzsch's theorem]]
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| :[[triangle-free graph|triangle-free]] planar graphs are 3-colorable.
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| ;[[Hadwiger–Nelson problem]]
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| :how many colors are needed to color the plane so that no two points at unit distance apart have the same color?
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| ;[[List of sets of four countries that border one another]]
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| :Contemporary examples of national maps requiring four colors
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| ;[[Apollonian network]]
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| :The planar graphs that require four colors and have exactly one four-coloring
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| ==Notes==
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| <references/>
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| ==References==
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| * {{Citation |last=Appel |first=Kenneth |last2=Haken |first2=Wolfgang |year=1977 |title=Every Planar Map is Four Colorable Part I. Discharging |periodical=Illinois Journal of Mathematics| volume=21 |pages=429–490| isbn= }}
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| ==External links==
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| {{commons category|Four color theorem}}
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| * {{springer|title=Four-colour problem|id=p/f040970}}
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| * {{MathWorld|title=Blanuša snarks|urlname=BlanusaSnarks}}
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| * {{MathWorld|title=Map coloring|urlname=MapColoring}}
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| [[Category:Graph coloring]]
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| [[Category:Planar graphs]]
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| [[Category:Theorems in graph theory]]
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| {{Link GA|zh}}
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| {{Link FA|zh}}
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