Four-color theorem

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Topic: Four-color theorem

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	Four color theorem - Wikipedia, the free encyclopedia
		The four color theorem was the first major theorem to be proved using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof).
		The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color.
		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.
	en.wikipedia.org /wiki/Four_color_theorem (2087 words)

	Ivars Peterson's MathLand
		The search for a simple, incisive proof of the four color theorem goes on, suggesting new puzzles and leading to novel mathematical techniques that turn out to be useful in applied mathematics and computer science.
		Four colors turn out to be necessary in any situation in which a region has common borders with an odd number of neighboring regions.
		The question is whether four colors are always enough to fill in every conceivable map that can be drawn on a flat piece of paper so that no countries sharing a common boundary are the same color.
	www.maa.org /mathland/mathland_1_6.html (1007 words)

	Rick Mabry
		Theorem A. Proposition A is equivalent to the Four Color Theorem.
		Theorem B. The Four Color Theorem is equivalent to the proposition that every planar triangulation with more than three vertices is the union of two connected bipartite graphs, each with no isthmus.
		It is not surprising that Hedetniemi did not obtain the result of Proposition A, since Proposition A is equivalent to the Four Color Theorem, the proof of which was announced in [1](1976).
	www.lsus.edu /sc/math/rmabry/bica/4color4web.htm (1053 words)

	Mappa.Mundi Magazine - Locus - The Four-Color Map Problem
		Some say the four color theorum was finally proved by Appel and Haken in 1976, but others claim that the question is yet to be resolved satisfactorily.
		It is possible to go to Four Corners and put one hand or foot in each of the states, but on a map two diagonally opposed states could be the same color.
		Bridgman's, 1896 Rail Road and Township map of New York illustrates the four color mapping problem - as a practical matter, green, yellow, pink and tan are sufficient to map the townships.
	mappa.mundi.net /locus/locus_014 (873 words)

	qg8.2
		This is one of the hardest theorems in all of mathematics.
		Theorem: For any trivalent planar graph without edge-loops, there exists a way to label the edges by the letters i, j and k so that no two edges meeting at a vertex are labelled by the same letter.
		He wondered whether it was always possible to color any map with only 4 colors, in such a way that no two countries (or counties!) touching with a common stretch of boundary were given the same color.
	math.ucr.edu /home/baez/qg-fall2000/qg8.2.html (2445 words)

	The Four Color Theorem
		In this context the four color theorem tells us that a space of four dimensions is sufficient to enable us to assign one of the four basis vectors to each vertex of a planar graph in such a way that the vectors of every pair of adjacent vertices are orthogonal.
		Also, Hamilton made contributions to graph theory (such as the idea of a Hamiltonian circuit, i.e., a path along the edges of a graph that visits each vertex exactly once), a subject that was developed largely through efforts to prove the four color conjecture.
		The coloring of geographical maps is essentially a topological problem, in the sense that it depends only on the connectivities between the countries, not on their specific shapes, sizes, or positions.
	www.mathpages.com /home/kmath266/kmath266.htm (4081 words)

	Four Colour Theorem
		A simple proof of this theorem has eluded mathematicians for centuries, but here is a proof that the four colour theorem is true, and that there is no planar map that needs 5 or more colours.
		The four colour theorem states that any planar (flat 2d) map needs only 4 colours to be coloured in.
		There are generally many alternatives using only 4 colours, and the Four Colour theorem asks 'Is there a map that we can create that requires 5 colours?'.
	www.users.globalnet.co.uk /~perry/maths/fourcolour/fourcolour.htm (658 words)

	Joseph Malkevitch: Four-Color Problem Tidbit
		For example, it is not difficult to "reduce" the general question of coloring the faces of plane maps with four or fewer colors to the problem of coloring the faces of a 3-valent plane map with four or fewer colors.
		First, instead of showing that the faces of every plane graph could be colored with four or fewer colors, the "dual problem" of coloring the vertices of a plane graph with four or fewer colors was studied.
		The need for a computer in the two proofs that have been found for the four-color problem is a result of the vast number of cases that must be considered in the interplay between discharging and the theory of unavoidable sets.
	www.york.cuny.edu /~malk/tidbits/tidbit-four-color.html (1418 words)

	Last doubts removed about the proof of the Four Color Theorem
		Guthrie's question became known as the Four Color Problem, and it grew to be the second most famous unsolved problem in mathematics after Fermat's last theorem.
		To prove the (network version of the) Four Color Theorem, you start out by assuming that there is a network that cannot be colored with four colors, and work to deduce a contradiction.
		The story of the Four Color Problem begins in October 1852, when Francis Guthrie, a young mathematics graduate from University College London, was coloring in a map showing the counties of England.
	www.maa.org /devlin/devlin_01_05.html (1481 words)

	Claim of Proof to Four-Color Theorem
		The nub of the proof lies in two theorems, the first essential to the method, and the second a crucial restatement of the fourcolour conjecture in a way that renders it susceptable to proof.
		At this stage we had little doubt that the four colour conjecture was true, ard various mathematicians with whom we corresponded, including Bertrand Russell, were aware that we had a method that was almost certainly capable of proving it.
		This elegant theorem I have now proved, and the proof will have been published by the time you are able to print this communication.
	www.lawsofform.org /gsb/nature.html (688 words)

	Four Colour Theorem
		A New Proof of The Four Color Theorem by Ashay Dharwadker.
		A NEW PROOF OF THE FOUR COLOUR THEOREM
	www.geocities.com /dharwadker (19 words)

	INRIA - A promising collaboration between INRIA and Microsoft Research
		The theorem asserts that four colors are enough to color any geographical map in such a way that no neighboring two countries are of the same color.
		This is thus the first time that the whole of the proof of the four color theorem is expressed in one and the same language, thus avoiding mixing disciplines-half mathematics, half computer science-a potential source of errors.
		It is the latter characteristic that makes such results as the four color theorem that cannot by described by a traditional mathematical text amenable for Coq.
	www.inria.fr /actualites/2005/theoreme4couleurs.en.html (766 words)

	Four Colour Theorem
		The Four Colour Theorem is the hypothesis that the maximum number of colours needed to fill the map is four.
		The Four Colour Theorem is to mathematicians what the song of the Sirens was to the sailors of ancient times.
		And while computer-aided proofs have begun to gain acceptance, largely thanks to the Four Colour Theorem, there remains the feeling that beauty, elegance and insight should triumph over the horror of a computer-generated proof.
	www.justinmullins.com /four_colour_theorem.htm (376 words)

	Four Color Theorem
		A summary of a new proof of the four color theorem is at Georgia Tech for which programs and data supplements are available by FTP.
		We take a pair of triangulations of a polygon and four color the vertices such that no two of the same color are connected by an edge of the triangulations.
		A survey (in PostScript) of the paper "The Four-Colour Theorem" is there also.
	grail.cba.csuohio.edu /~somos/4ct.html (329 words)

	The Four Color Problem
		Be careful not to confuse the four-color theorem with graph coloring problems involving the coloring of vertices as opposed to regions.
		The four color theorem applies only to planar (or spherical) "maps", not to regions drawn on other surfaces.
		Mathematics journals regularly received, and discarded (probably unread) supposed "proofs" of the Four-Color Theorem.
	bhs.broo.k12.wv.us /discrete/4Color.htm (578 words)

	Serendip: The 4 color problem
		Four-Color Theorem, from Eric Weisstein's Treasure Trove of Mathematics
		The "four color problem" is a simple and yet quite significant problem in mathematics, with implications for thinking about human understanding generally.
		Some of these history, significance and implications of the four color problem will be explored in the present exhibit, which is currently under development.
	serendip.brynmawr.edu /playground/fourcolor (292 words)

	ryanp.htm
		The concept of the Four Coloring Theorem was born in 1852 when Francis Guthrie noticed that he only needed four different colors to color in a map of England.
		I find the Four Coloring Theorem to be very interesting because of it's apparent simplicity paired with it's long, laborious struggle to be proved.
		To prove the Six Color Theorem we began by making a Lemma which states that, "At least one vertex of a planar graph has order five or less." The order of a vertex is the number of edges coming out of that vertex.
	www.facstaff.bucknell.edu /udaepp/090/w3/ryanp.htm (1535 words)

	Four Color Theorem
		The Four Color Theorem was solved by Haken and Appel in 1976, with a proof that involved the use of computers.
		Howard Levi, at the end of a long and distinguished career in mathematics, spent the last years of his life working on a proof of the Four Color Theorem along algebraic lines.
		Coloring and orientations of graphs, Combinatorica, 12, 1992, 125-134.
	comet.lehman.cuny.edu /fitting/fourcolor/fourcolor.html (362 words)

	Seymour is Solved!!!
		Four Color Theorem by Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas.
		The Four Color Theorem states that any map can be colored using four colors in such a way that adjacent regions (i.e.
		It took less than four hours of CPU time to complete the proof of the Four Color Theorem.
	www-unix.mcs.anl.gov /metaneos/seymour (485 words)

	No. 1961: The Four-Color Problem
		The seeming simplicity of the four color problem led countless people to try their hand at it over the years, including some of the world's most renowned mathematicians.
		But debate about the proof of the four color problem lingers.
		Unlike many mathematical problems that rely on weaving together a small number of ideas, all the attempted proofs of the four color problem reduce to checking many, many specially constructed maps.
	www.uh.edu /admin/engines/epi1961.htm (680 words)

	The Four Color Theorem
		The four color theorem states that the chromatic number of a planar graph is no greater than four.
		In my presentation I will explain the four color theorem and its history.
	www.usip.edu /research/scholarlyday/abstractDetail.asp?id=148 (31 words)

	An Intuitive Proof of the Four Color Theorem
		The four color theorem essentially says that any map can have each country colored once by one of four colors without any adjacent countries sharing a color.
		An Intuitive Proof of the Four Color Theorem
		All triangle meshes created by subdividing a face are three-colorable, therefore the countries being connected to in the symmetric graph are three-colorable, therefore the symmetric graph is four-colorable, therefore any instance of an inside insert or outside insert is four-colorable, therefore all maps are four-colorable.
	www.rpi.edu /~mcdonk/writing/4ct (889 words)

	TB Stumper Answers: 15 December 2000 - Four-Color Christmas
		That theorem, as all readers of this department must know, is that four colors are both necessary and sufficient for coloring all planar maps so that no two regions with a common boundary are the same color.
		A new proof of the Four Color Theorem was announced in 1996 by Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas.
		The four-color theorem was "proved" (or verified?) in 1976 by Kenneth Appel and Wolfgang Haken of the University of Illinois.
	www.rain.org /~mkummel/stumpers/15dec00a.html (1545 words)

	MathForge.net--Power Tools for Online Mathematics
		Reader Proposes New Proof of Four Color Theorem
		The original proof of the Four Color Theorem has been formalized in Coq.
		Yes you can color G' with 4 or less colors, but in order for the created regions to collapse back to a single point requires that all the points on that circle be the same color, which you did not prove to be possible.
	mathforge.net /index.jsp?page=seeReplies&messageNum=2413 (924 words)

	The Four Color Problem
		The four-color theorem is one of the most famous problems in mathematics.
		But that is not the end of the story: As Wilson pointed out, the four-color theorem is really just a special case of more general mathematical questions waiting to be answered.
		It says that, given any map on a flat plane, at most four colors are needed to color the different regions of the map in such a way that no two adjacent regions have the same color.
	www.ams.org /ams/wilson-jmm2003.html (304 words)

	Ideas, Concepts, and Definitions
		Mapmakers are not mathematicians, so the assertion that only four colors would be necessary for all maps gained acceptance in the map-making community over the years because no one ever stumbled upon a map that required the use of five colors.
		The Four Color Problem was famous and unsolved for many years.
		The basic rule for coloring a map is that no two regions that share a boundary can be the same color.
	www.cs.uidaho.edu /~casey931/mega-math/gloss/math/4ct.html (541 words)

	Four Color Theorem - Uncyclopedia
		It is a controversial theorem because its proof was carried out in large parts by computers, which automatically colored in a map of the entire world in four colors.
		The Four Color Theorem states that any map can be colored in using not more than four different colors, namely: blue, for water; green, for land; yellow, for desert; and red, to mark branches of Little Chef.
		The computer which solved the Four Color Theorem is called Deep Blue.
	uncyclopedia.org /wiki/Four_Color_Theorem (199 words)

	Proof of four color theorem proved
		The four color theorem states that any flat map can be colored in a minimum of four colors where no two regions sharing a non-zero length boundary have the same color.
		reported that a 1976 proof of the famous four color theorem that involved a complicated computer program has itself been proved correct by translating the proof into a language where the steps of the proof can be checked by "logic-checking software".
		Suppose that the researchers did achieve a translated program that is in fact a verifiable proof of the four color theorem.
	homepage.mac.com /duanewilliams/iblog/C2129834616/E1316939883 (146 words)

	Dharwadker's Alleged Proof at Wikipedia
		In which case you are again claiming that the four color theorem is false and that your specific map is a counter-example to the four color theorem!
		You must prove that it is a counter-example to the four color theorem (if you want to ensure that no two regions will be identified during your construction).
		In my opinion, Dharwadker's proof is correct, so the four color theorem is true and so any map on the plane can be properly colored with at most 4 colors.
	www.log24.com /theory/Dharwadker/Wiki.html (7380 words)

	Georges Gonthier's HomePage
		Users of the Coq system, or brave souls wishing to delve into the actual Coq Proof of the Four Color Theorem, might also be interested in a description of the notations and the proof command language that were used to write the proof.
		The latter subject inspired his latest work, a fully checked formal proof of the famous Four Colour Theorem, using the Coq proof assistant developed at INRIA, the French Institut National de Recherche en Informatique et Automatique (this was work in collaboration with Benjamin Werner of INRIA).
		Mathematically inclined readers may consult a paper describing the mathematics of the formalization.
	research.microsoft.com /~gonthier (141 words)

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