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Topic: Four color problem

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Four color theorem

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	Four color theorem - Wikipedia, the free encyclopedia
		It is obvious that three colors are inadequate: this applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.
		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.
		The problem on the sphere is equivalent to that on the plane.
	en.wikipedia.org /wiki/Four_color_theorem (1840 words)

	Four color theorem -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-10)
		The four color theorem states that every possible geographical map can be colored using no more than four (A visual attribute of things that results from the light they emit or transmit or reflect) colors in such a way that no two adjacent regions receive the same color.
		It is obvious that three colors are inadequate, and it is not difficult, relatively speaking, to prove that five colors are sufficient to color a map.
		The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be unfeasible for a human to verify by hand.
	www.absoluteastronomy.com /encyclopedia/f/fo/four_color_theorem.htm (1767 words)

	Mappa.Mundi Magazine - Locus - The Four-Color Map Problem
		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.
		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.
		The four-color map problem was first proposed in 1852 by Francis Guthrie, a mathematician who was engaged in drawing a map of the counties of Britain.
	mappa.mundi.net /locus/locus_014 (873 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.
		For a vertex coloring of a graph (one can color the vertices of any graph while it is meaningful to discuss face coloring problems only for plane graphs) one assigns a label to each vertex so that vertices joined by an edge get different labels.
	www.york.cuny.edu /~malk/tidbits/tidbit-four-color.html (1418 words)

	Last doubts removed about the proof of the Four Color Theorem
		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.
		The coloring has to meet the obvious requirement that no two regions (countries, counties, or whatever) sharing a length of common boundary should be given the same color.
		Indeed, the problem of coloring the map (in the sense of Guthrie's problem) can be reformulated in terms of coloring the network: color the nodes of the network in such a way that any two nodes which are connected together must have different colors.
	www.maa.org /devlin/devlin_01_05.html (1481 words)

	The Four Color Theorem (Site not responding. Last check: 2007-10-10)
		The Four Color Problem dates back to 1852 when Francis Guthrie, while trying to color the map of counties of England noticed that four colors sufficed.
		This was confirmed by Appel and Haken in 1976, when they published their proof of the Four Color Theorem [1,2].
		A. Kempe, On the geographical problem of the four colors, Amer.
	www.math.gatech.edu /~thomas/FC/fourcolor.html (1878 words)

	Archimedes Plutonium (Site not responding. Last check: 2007-10-10)
		The 4 color problem of mathematics is a vivid example of where mathematicians create a problem due to their inability to "well define" the issues.
		In a sentence the four color mapping problem reduces to 2 color mapping because there are two main entities-- border lines and interiors.
		So, you need two colors, one color for border lines and I always prefer fl or blue ink and one color for land interior which I prefer to be white.
	www.iw.net /~a_plutonium/File111.html (2267 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-10)
		The Four Color Problem is one of my favorites because it is the first unsolved (it was at the time) problem that I was ever introduced to.
		To color a map, you must assign each region a color and no two regions may have the same color if they share a side (one point doesn't count).
		Date: Thu, 8 Dec 1994 22:03:40 -0500 From: Stephen Weimar Subject: Re: Four color Map Problem The four-color map theorem was proved by Appel and Haken at the University of Illinois at Urbana-Champaign in 1976.
	mathforum.org /library/drmath/view/57231.html (392 words)

	Learn more about Four color theorem in the online encyclopedia. (Site not responding. Last check: 2007-10-10)
		The four color theorem was the first major theorem to be proven using a computer, and the proof was not accepted by all mathematicians because it could not directly be verified by a human.
		Under these conditions, the corresponding graph need no longer be planar, and four colors are not always sufficient.
		Five colors are required if those two regions are to receive the same color.
	www.onlineencyclopedia.org /f/fo/four_color_theorem.html (682 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		The Four Color Theorem was originally a conjecture that, for a given planar map of regions, no more than four colors were required to color each region in such a way that it would be distinct from its adjacent regions.
		So, we go from "no more than four colors are required to color the map" to "it is not the case that a region is adjacent to four differently colored regions".
		The reason four regions adjacent to a given region would have four different colors is because each of those regions is also adjacent to at least three differently colored regions, etc. With a little thought, you should see that the adjacency relationship is the key.
	www.klbrun.com /philosophy/proof.html (517 words)

	Read This: Four Colors Suffice: How the Map Problem Was Solved
		The ring of states around Nevada cannot be colored with two colors (two adjacent states would have to be the same color), so it takes three colors to color this ring of states.
		The analogous problem for the torus is not difficult: every map on a torus can be colored with seven colors and there are such maps that require seven colors.
		It was Birkhoff's contribution that led to the mention in the preface of coloring maps on a honeymoon.
	www.maa.org /reviews/fourcolors.html (1972 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-10)
		He was colouring in a map of the counties of Britain and was intrigued by the fact that four colours appeared to be sufficient regardless of the complexity of boundary shapes or how many regions had a commom border.
		The first breakthrough in the four-colour problem came in 1922 when Philip Franklin ignored the general problem and settled for a proof which showed that any map containing 25 or fewer regions required only four colours.
		In 1975, after five years of working on the problem, they turned to the new number-cruncher and in 1976 after 1200 hours of computer time were able to announce that all 1482 maps had been analysed and none of them required more than four colours.
	mathforum.org /library/drmath/view/52535.html (528 words)

	The Four Color Problem (Site not responding. Last check: 2007-10-10)
		The problem was often trotted out in mathematics classroom both to provide a diversion and to prove to students that mathematics still had some unanswered questions whose statements were within their capabilities to understand, if not solve.
		The problem was finally solved in 1976 by Kenneth Appel and W. Haken, of the University of Illinois, using a second generation Cray computer to analyze 1900 possible arrangements of regions in a plane.
		Be careful not to confuse the four-color theorem with graph coloring problems involving the coloring of vertices as opposed to regions.
	bhs.broo.k12.wv.us /discrete/4Color.htm (578 words)

	Four color theorem - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-10)
		The four color theorem states that every possible geographical map can be colored with at most four colors in such a way that no two adjacent regions receive the same color.
		It is obvious that three colors are completely inadequate, and it is not difficult to prove that five colors are sufficient to color a map.
		Significant results in that area were produced by Croatian mathematician Danilo Blanuša; in the 1940s by finding an original snark.
	encyclopedia.learnthis.info /f/fo/four_color_theorem.html (674 words)

	The four colour theorem
		Charles Peirce in the USA attempted to prove the Conjecture in the 1860's and he was to retain a lifelong interest in the problem.
		Reynolds increased it to 27 in 1926, Winn to 35 in 1940, Ore and Stemple to 39 in 1970 and Mayer to 95 in 1976.
		The Four Colour Theorem was the first major theorem to be proved using a computer, having a proof that could not be verified directly by other mathematicians.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/The_four_colour_theorem.html (1691 words)

	Atlas: Newton, Klein, Kauffman, Cayley and the Four Color Problem by Paul C. Kainen (Site not responding. Last check: 2007-10-10)
		An argument is given to show that the problem of map coloring in the plane, well-known to be equivalent to a problem involving pairs of rooted, cubic, plane trees, may be regarded as a generalization of Newtonian mechanics.
		Coloring is equivalent to the propagation of torque.
		Since it was Cayley who first announced the Four Color Problem to the mathematical community, it is ironic that the ``numbers'' he invented (with Graves) may be required for a natural solution.
	atlas-conferences.com /cgi-bin/abstract/cafy-06 (298 words)

	[No title]
		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.
		The Four Color Theorem was the first major theorem to be proved using a computer.
		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)

	[No title]
		Thus, it requires an odd number of step to walk through all the bridges of a place but that require the person to stay in the node finally if he/she were originally from the outside and to be outside if he/she were originally from the inside.
		Four Color Problem What is the smallest number of colors needed to color any planar map so that any two neighboring regions have different colors?
		Hamilton was evidently too interested in other things to work on the four color problem, and it lay dormant for about 25 years.
	web.mit.edu /chungc/urop02/GraphTheory (2652 words)

	chris
		Another problem to be considered is the proof of the Five-color theorem.
		Furthermore, when we begin reinstalling vertices and edges after the removal process, we still need to be able to color the graph with four colors; by this we mean that the reintroduced vertex cannot have a similar edge with at least one of the original four vertices.
		I think that these problems are good because they seem so easy at first, but once you get into the problem, it’s more than just coloring.
	www.facstaff.bucknell.edu /udaepp/090/w3/chrisc.htm (1822 words)

	American Scientist Online - Map Quest (Site not responding. Last check: 2007-10-10)
		The "fact" is that only four colors are required to color any map in such a way that adjacent regions receive different colors.
		Francis decided that it must be true after coloring a map of the counties in England, and he allowed Frederick to submit the challenge to De Morgan.
		In the final chapter of Four Colors Suffice, Wilson surveys the skeptical reactions of the mathematics community to the use of this strategy.
	www.americanscientist.org /template/AssetDetail/assetid/21979 (848 words)

	Some Probabilistic Restatements of the Four Color Conjecture (ResearchIndex) (Site not responding. Last check: 2007-10-10)
		The Four Color Conjecture turns out to be equivalent to di#erent statements about positive correlation among some pairs of these events.
		24 The four-color problem (context) - Ore - 1967
		5 The four-color problem (context) - Saaty, Kainen - 1977
	citeseer.ist.psu.edu /matiyasevich03some.html (431 words)

	The Four Color Problem and its connection to South African Flora (Site not responding. Last check: 2007-10-10)
		In the 1850's Francis Guthrie was the first mathematician to formulate the Four Color Problem.
		Awareness of the Four Color Problem increased substantially when, on June 13, 1878, the renowned mathematician Arthur Cayley asked if the problem had been solved.
		Shortly afterwards, Cayley [2] published a paper on the Four Color Problem, in which he postulated why this problem appears to be so difficult.
	io.uwinnipeg.ca /~ooellerm/guthrie/FourColor.html (703 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.
		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.
		In 1996, four researchers set out to verify the Appel-Haken proof because, in their own words, "as far as we know, no one has yet verified it in its entirety.
	www.uh.edu /engines/epi1961.htm (680 words)

	Serendip: The 4 color problem (Site not responding. Last check: 2007-10-10)
		The "four color problem" is a simple and yet quite significant problem in mathematics, with implications for thinking about human understanding generally.
		The Four Color Problem and its Connection to South Arican Flora, by Ostrid Oellermann, University of Winnipeg
		The question is how many colors are required to color such a map, using the rule that no two countries with adjacent borders may be the same color (countries meeting at a point may be the same color).
	serendip.brynmawr.edu /playground/fourcolor (292 words)

	Powell's Books - Four Colors Suffice: How the Map Problem Was Solved by Robin Wilson
		The problem posed in the letter came from a former student: What is the least possible number of colors needed to fill in any map (real or invented) so that neighboring countries are always colored differently?
		But the problem set off a frenzy among professional mathematicians and amateur problem-solvers, among them Lewis Carroll, an astronomer, a botanist, an obsessive golfist, the Bishop of London, a man who set his watch only once a year, a California traffic cop, and a bridegroom who spent his honeymoon coloring maps.
		This is the entertaining story of those who failed to prove, and those who ultimately did prove, that four colors do indeed suffice to color any map.
	www.powells.com /cgi-bin/biblio?isbn=0691115338 (458 words)

	Toward an Inductive Solution for the 4 Color Problem
		Inductive arguments to prove the 4 color theorem have failed in the past because no method was found for generating all maximal planar graphs inductively.
		The inductive approach assumes that a maximal graph of size N (or less) is colorable; it then shows that any means of generating a maximal planar graph of size N+1 from one of size N (or less) produces one that is still colorable.
		In order for V' and V" to have different colors, the dividing point of the K-circuit, the ends of the edges that were split, must divide the K circuit in such a way that any new 3-coloring of each division differs in the third color.
	www.xenodochy.org /article/fourcolor.html (1978 words)

	The Four
		Now mathematicians know that only four colors are needed to make a map where none of the countries that touch another are the same color.
		Guthrie was a part-time mathematician who, one day, while drawing a map of the counties of Britain noticed he needed only four colors to color the map with no county touching others with its same color.
		A map only needs four colors to be colored with no two regions that are adjacent the same color.
	www.science.gmu.edu /~ssmith7/MapPage.htm (493 words)

	Colorful Mathematics/Games/4colors (Site not responding. Last check: 2007-10-10)
		This game is based on a simple idea: draw a map or any picture as complicated as you wish and color each region using the fewest possible number of colors, the only requirement being that regions sharing a common border must receive different colors.
		It took over a hundred years for mathematicians to prove that four colors were sufficient, no matter how complicated the map.
		We believe however, that a truer feeling for the problem can be obtained only by drawing the maps themselves, testing ideas, and several drawing tools are provided for that purpose.
	www.math.ucalgary.ca /~laf/colorful/4colors.html (417 words)

	No. 1919: Möbius
		When he was fifty, Möbius gave a lecture in which he posed an odd problem: You're the king, and you must divide your kingdom among your five sons.
		A somewhat similar problem is proving that you can color any map using only four colors.
		The map problem is not really the same as the kingdom problem, but Möbius gets wide credit for inventing that four-color problem.
	www.uh.edu /engines/epi1919.htm (644 words)

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