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Topic: Four color theorem

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	Math Forum - Ask Dr. Math Archives: Four-Color Map Theorem
		I hear the Four-color map theorem was either proved or disproved and that extensive computer effort was required....
		An extension of the four-color map theorem to the mobius strip, i.e.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /library/drmath/sets/select/dm_4-color.html (218 words)

	Andrew Yang
		The theorem does not state that four colors are always necessary; in special cases three, or even two colors may suffice.
		The Four Color Theorem made its appearance in the mathematical community in 1852, when an undergraduate student in London, Francis Guthrie, asked his older brother Frederick via a letter whether it was possible to color any map using only four colors so that adjacent regions would be different colors.
		Four colors are necessary to color this map, because each region is adjacent to three other regions.
	www.its.caltech.edu /~sciwrite/journal03/yang.html (3235 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.
		Like Fermat's last theorem, there are some "obvious" ways to solve the problem that seem, on the face of it, to work, but have subtle errors, and professional mathematicians grew used to receiving claimed proofs from amateurs who would often remain convinced their solution was correct even after the error was pointed out to them.
	www.maa.org /devlin/devlin_01_05.html (1470 words)

	Ivars Peterson's MathTrek
		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.
		Because the conjecture that four colors suffice hadn't yet been proved, Carroll didn't know with certainty whether the answer to his question was four or five.
		Four colors turn out to be necessary in any situation in which a region has common borders with an odd number of neighboring regions.
	www.maa.org /mathland/mathland_1_6.html (926 words)

	Mappa.Mundi Magazine - Locus - The Four-Color Map Problem
		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.
		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.
	mappa.mundi.net /locus/locus_014 (873 words)

	four-color - Search Results - MSN Encarta
		Color, physical phenomenon of light or visual perception associated with the various wavelengths in the visible portion of the electromagnetic...
		The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the states of a country, the...
		Computer aided proof of the four color theorem by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas.
	encarta.msn.com /four-color.html (248 words)

	Four color theorem Summary
		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.
		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).
		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.
	www.bookrags.com /Four_color_theorem (2866 words)

	Claim of Proof to Four-Color Theorem
		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.
		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.
		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)

	Math G Mission College Santa Clara
		He demonstrated that every map can be five colored as well as proving that if the number of edges surrounding each region is divisible by three, then the regions all require a maximum of four colors.
		One of the concepts used in the development of proof for the Four Colour Theorem was the use of the ìgreedyî or ìsingle mindedî algorithm.
		The Four Color Theorem is the first proof that used a computer and was not actually able to be verified by hand.
	www.missioncollege.org /Depts/Math/beard2.htm (2429 words)

	four-color problem
		A long-standing problem that dates back to 1852 when Francis Guthrie, while trying to color a map of the counties of England noticed that four colors were enough to ensure that no adjacent counties were colored the same.
		While the concept of reducibility was studied by other researchers as well, it seems that the idea of discharging, crucial for the unavoidability part of the proof, is due to Heesch, and that it was he who conjectured that a suitable development of this method would solve the Four-Color Problem.
		The Four-Color theorem is true for maps on a plane or on a sphere.
	www.daviddarling.info /encyclopedia/F/four-color_problem.html (616 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.
	york.cuny.edu /~malk/tidbits/tidbit-four-color.html (1418 words)

	qg8.2
		This is one of the hardest theorems in all of mathematics.
		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.
		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.
	math.ucr.edu /home/baez/qg-fall2000/qg8.2.html (2445 words)

	IMAGE\SOLSTICE\WIN98\4color (Site not responding. Last check: )
		In the coloring scheme below, red was generally used as first choice, green as second, yellow as third, and purple as fourth.
		On occasion, the general strategy was violated in order to color efficiently; for example, Montana was colored green so that Idaho could be colored red in a vertical alternation pattern of red/green/red.
		Surprisingly, the solution to coloring requirements on surfaces other the plane were determined well ahead of the solution in the plane.
	www-personal.umich.edu /~copyrght/image/solstice/win98/4color.html (262 words)

	The four colour theorem
		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 year 1976 saw a complete solution to the Four Colour Conjecture when it was to become the Four Colour Theorem for the second, and last, time.
		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 (1703 words)

	Colorful Mathematics/Teacher
		However, using only four colors is not as straightforward as it may appear so six colors have been provided which should make it relatively easy for everyone to color their maps correctly.
		After having completely colored a map, a message will appear on the screen asking the student to try to improve on the number of colors required, or to be able to show that this number cannot be reduced.
		The use of one color to select some vertices in a graph so that every vertex is either colored or connected by an edge to a colored one creates a dominating set of a graph.
	www.math.ucalgary.ca /~laf/colorful/teacher.html (2018 words)

	Math Pioneers
		In fact, four colors are enough to distinguish every community from its neighbors on any map that has ever been made or ever could be made, no matter how convoluted or weird-looking.
		They found that four colors were always enough, but mathematicians, who tend to take ideas to extremes, wondered whether four would always be enough.
		As it turned out, in the original proof of the four-color theorem, the fact that almost 2,000 particular configurations could not appear in a smallest map needed to be verified by computer.
	www.unhmagazine.unh.edu /sp02/mathpioneers.html (924 words)

	Interactivate: Activities
		Color numbers in Pascal's Triangle by rolling a number and then clicking on all entries that are multiples of the number rolled, thereby practicing multiplication tables, investigating number patterns, and investigating fractal patterns.
		Color numbers in Pascal's Triangle by rolling a number and then clicking on all entries that have the same remainder when divided by the number rolled, thereby practicing division and remainders, investigating number patterns, and investigating fractal patterns.
		Visualize fractions by coloring in the appropriate portions of either a circle or a square, then order those fractions from least to greatest.
	www.shodor.org /interactivate/activities (3137 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)

	[No title]
		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.
		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.
		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)

	The Four Color Theorem
		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.
		Likewise the four vectors assigned to the vertices of a tetrahedral graph would be a complete set of four mutually orthogonal vectors (a tetrad), but still with arbitrary orientation.
		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.
	www.mathpages.com /home/kmath266/kmath266.htm (4081 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 (1975 words)

	The Four Color Theorem
		This page gives a brief summary of a new proof of the Four Color Theorem and a four-coloring algorithm found by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas.
		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].
	www.math.gatech.edu /~thomas/FC/fourcolor.html (1895 words)

	Four Colour Theorem: A Small Historical Insight.
		A coloring of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent vertices are assigned the same color.
		The Four Color Theorem is shrouded in confusion and controversy.
		They revealed that if the Four Color Theorem were false, there would have to be a counterexample in a set of approximately 2000 different types of counterexamples, and that none of these types actually exists.
	student.adams.edu /~verderaimedj/finalEssay (1552 words)

	Notes on Finite Geometry (Site Map)
		Research announcement (4x4 case of diamond theorem and algebraic generalization) This research announcement was the basis for an abstract (79T-A37) in the Feb. 1979 AMS Notices.
		An invariance of symmetry The diamond theorem on a 4x4x4 cube, and a sketch of the proof.
		Portrait of O A table of the octahedral group O using the 24 patterns from the 2x2 case of the diamond theorem.
	finitegeometry.org /sc/map.html (794 words)

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