61. A New Proof Of The Four-Colour Theorem The fourcolour theorem, that every loopless planar graph admits a vertex-colouringwith at most four different colours, was proved in 1976 by Appel and http://www.mpim-bonn.mpg.de/external-documentation/era-mirror/1996-01-003/1996-0

This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ A new proof of the four-colour theorem Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas Abstract. The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we announce another proof, still using a computer, but simpler than Appel and Haken's in several respects. Retrieve entire article TeX source DVI PostScript Article Info ERA Amer. Math. Soc. (1996), pp. 17-25 Publisher Identifier: S 1079-6762(96)00003-5 Mathematics Subject Classification . Primary 05C10, 05C15, 05C85 Received by the editors September 26, 1995 Communicated by Ronald Graham Comments (When Available) Neil Robertson Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus, Ohio 43210, USA E-mail address: robertso@math.ohio-state.edu Daniel P. Sanders School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA E-mail address: dsanders@math.ohio-state.edu

62. Oxford University Press: Graphs, Colourings And The Four-Colour Theorem: Robert The fourcolour theorem is one of the famous problems of mathematics, In PartIII we return to the four-colour theorem, and study in detail the methods http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?ci=01

63. Oxford University Press: Graphs, Colourings And The Four-Colour Theorem: Robert The fourcolour theorem is one of the famous problems of mathematics, In PartII we return to the four-colour theorem, and study in detail the methods http://www.oup.com/us/catalog/general/subject/Mathematics/PureMathematics/?ci=01

64. 4-colour Theorem Apparently the artist did not realize that four colors would have sufficed. An Update on the fourColor theorem, Notices of the Amer. Math. Soc. http://mathcentral.uregina.ca/RR/database/RR.09.97/fisher1.html

The 4-Colour Theorem by Chris Fisher Department of Mathematics and Statistics University of Regina A nice discussion of map coloring can be found in "The Mathematics of Map Coloring," which Professor H.S.M. Coxeter wrote for the Journal of Recreational Mathematics, 2:1 (1969). He began by pointing out that in almost any atlas, 5 or 6 colors are used in a map of the United States to distinguish neighboring states. "Apparently the artist did not realize that four colors would have sufficed. (It is understood that two states may be colored alike if they merely have a point in common, as in the case of Arizona and Colorado.)" This leads to the mathematical question, Can every conceivable map (on a sphere or a plane) be colored with four colors, or does some particular map really need five? The question was first posed in 1852 by Francis Guthrie, a mathematics graduate student in London at the time. He had noticed the sufficiency of four colors for distinguishing the counties in a map of England. The question was passed along to several important British mathematicians (De Morgan, Hamilton), but apparently it was not seriously investigated until Cayley in 1878 challenged the members of the London Mathematical Society to solve it. From that time until its computer solution nearly 100 years later the problem stood alongside Fermat's last theorem among the great mathematical challenge of the century. Like the Fermat problem, the map-coloring question is easily stated and can easily be understood by anybody. Both problems lack any important consequences, yet have led to extraordinarily important new mathematical ideas and techniques. Both problems are alluring and elusive.

65. The Map Room Archives (November 2003) The fourcolour theorem states that in order to have a map in which no twocountries that touch use the same colour, you only need to have a minimum of four http://www.mcwetboy.net/maproom/2003_11_01_archive.phtml

Read Hurricane Katrina Entries Home About Archives ... Suggest Link Archives (November 2003) Sunday, November 30, 2003 Warblogosphere Speaking of the blogosphere, says it all about this putative map of the blogosphere Posted by Jonathan Crowe at 8:09 PM Moscow Metro Just in case you were thinking that the London Underground was the only subway system getting any attention in the blogosphere, The Cartoonist links to the map of the Moscow Metro . In Russian, but it looks official. Posted by Jonathan Crowe at 8:04 PM Ancient Maps of Jerusalem Iconomy is posting again. Huzzah! Because one of her new links is to a site titled Ancient Maps of Jerusalem Posted by Jonathan Crowe at 7:59 PM Saturday, November 29, 2003 Guantanamo Bay Plep links to Eyeballing the Naval Station Guantanamo Bay a larger site that makes a point of shining light on sensitive places , be it military bases, nuclear facilities or the homes of politicians, in a sort of who-watches-the-watchers idiom. Controversial, especially if you were, say, working for the FBI Posted by Jonathan Crowe at 8:41 AM 3D London Underground The Web is unwilling to leave the London Underground alone.

66. Joseph Malkevitch: Four-Color Problem Tidbit fourColor theorem Tidbit (09/13/2003) Prepared by Joseph Malkevitch Wilson,R., Graphs colourings and the four-colour theorem, Oxford, Oxford, 2002. http://www.york.cuny.edu/~malk/tidbits/tidbit-four-color.html

Four-Color Theorem Tidbit (09/13/2003) Prepared by: Joseph Malkevitch Mathematics and Computing Department York College (CUNY) Jamaica, NY 11451 A common way to assist people get a feel for world geography is to display the various countries on a (spherical) globe. To help with understanding which countries share a border it is useful to have countries on the globe which share a common boundary (an edge, in graph theory terminology) have different colors. What is the minimum number of colors necessary to color the countries on a sphere so that countries that meet along an edge get different colors? To answer this mathematical question for spheres is a bit of a nuisance because as one looks at a globe one can only see the countries on one hemisphere, because the other countries are hidden from view. Is there some way of displaying the information on a sphere in a more convenient way? In fact there is! The problem of coloring maps on a sphere can be reduced to the problem of coloring maps in the plane by a simple geometrical transformation. This approach to problem solving is typical of the way that mathematicians work. Taking a problem in one setting and trying to recast into an easier related setting. The transformation this time is know as stereographic projection. Take a sphere and let N represent its North pole and S represent its South pole. Now let P be a plane tangent (i.e. touching at exactly one point) to the sphere at S. If R is any point of the sphere except N, the image of R in P, which will be denoted s(R) will be the point where the line NR meets the plane P. The two dimensional analog of stereographic projection, which should make clear what happens in 3-dimensions, is illustrated in the diagram below:. Notice that points which are close to the North pole are transformed to points which are very far away from the South pole.

67. Wu :: Forums - The Infinite Four-color Theorem Therefore the fourcolor theorem would hold for infinite planar graphs, which issame In fact, the four-colour theorem (4CT) is stated not in terms of http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_hard;action

68. Think Again! - Comments This got me wondering what the status of the fourcolour theorem (conjecture? Kempe thought he d proved the four-colour theorem in 1879. http://simpler-solutions.net/pmachinefree/thinkagain/comments.php?id=456_0_3_0_C

69. Problem 2.1 DP Sanders, P. Seymour, and R. Thomas The fourcolour theorem , Manuscript.Abstract The four-colour theorem, that every loopless planar graph admits http://www.imada.sdu.dk/Research/Graphcol/2.1.html

Problem 2.1: Four-Color Theorem N. Robertson, D.P. Sanders, P. Seymour, and R. Thomas "The Four-Colour Theorem", Manuscript. Abstract: "The four-colour theorem, that every loopless planar graph admits a vertex-colouring with at most four different colours, was proved in 1976 by Appel and Haken, using a computer. Here we give another proof, still using a computer, but simpler in several respects than Appel and Haken's." ( A copy of the manuscript is available from the authors. Computer files supporting the proof can be obtained via anonymous ftp - login as "anonymous" and give your e-mail address as password - from ftp.math.gatech.edu located in the directory pub/users/thomas/four ) An interesting and very well presented summary of the new proof can be found on the WWW under the address: http://www.math.gatech.edu/~thomas/FC/fourcolor.html A history of the four-color theorem can be found on the WWW under the address: http://www-groups.dcs.st-and.ac.uk:80/~history/HistTopics/The_four_colour_theorem.html December, 1994 Bjarne Toft The problem of deciding 3-colourability of a 4-regular graph mentioned on pages 34-35 was proved NP-complete already in 1980 in the paper: D.P. Dailey, Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete

70. The Four Color Problem And Its Connection To South African Flora Shortly afterwards, Cayley 2 published a paper on the four Color Problem, Soc. 1 (1879), 259261. PJ Heawood, Map-colour theorem. Quart. J. Math. http://io.uwinnipeg.ca/~ooellerm/guthrie/FourColor.html

The Four Color Problem and its connection to South African Flora In the 1850's Francis Guthrie was the first mathematician to formulate the Four Color Problem . He asked whether it is possible to color any map with four or fewer colors so that adjacent regions (those that share a common boundary) are colored differently. At the time when he posed the problem, he was a student at University College in London. He attempted to prove that the counties of any map could be colored in this map with four colors. However, he was not entirely satisfied with his proof, so he mentioned his problem to his brother Frederick, who, in turn, mentioned it to his instructor, the famous Augustus De Morgan (after whom De Morgan's Laws of set theory are named). In a letter dated October 23, 1852, De Morgan mentioned the problem to Sir William Rowan Hamilton (for whom hamiltonian graphs are named). In his response, Hamilton, perhaps displaying his insight into the difficulty of mathematical problems, replied to De Morgan that he did not plan to consider this problem in the near future. Evidently, De Morgan spoke often of this problem with other mathematicians. Indeed, De Morgan is credited with writing an anonymous article in the April 14, 1860, issue of the journal Athenaeum in which he discusses the Four Colour Problem. This is the first known published reference to the problem.

71. Graphs, Colourings And The 4CT Graphs, colourings and the fourcolour theorem 38 the proof of theorem 4.11is incomplete, as it does not deal with the case when the collapsing http://www.maths.qmul.ac.uk/~raw/graph.html

Graphs, colourings and the four-colour theorem Oxford University Press Corrections The following errors and misprints have already been found. If you have found any others, please email me: R.A.Wilson [ at qmul.ac.uk] p. 29: in line 1, 'negative' should read 'non-positive', and in Lemma 3.17 and Proposition 3.18 the summation should start with i=0. p. 38: the proof of Theorem 4.11 is incomplete, as it does not deal with the case when the collapsing produces a multigraph which is not a graph. p. 57: in Exercise 5.10, Delta = Delta(G). p. 61: the proof of Corollary 6.2 is inaccurate, as the case p-q+r=1 may also occur. It is better to continue adding edges until the faces are 2-cells, when p-q+r=0. p. 120: the Kempe-chain arguments in the proof of Theorem 10.9 are wrong. In case 4, either there is a blue-green chain from v to v , in which case we can change v to yellow; or there is a blue-green chain from v to v , in which case we can change v to yellow; or there is neither, in which case we can change v to green. In all three cases, we can now complete the colouring. Similarly, in case 5, either there is a blue-yellow chain from v

72. Abstract For Talk: Robertson Colloquium Talk The fourcolour theorem. Abstract for the Colloquium April 22, 1999 at 430.That planar maps can be properly colored in four colors was conjectured in 1852 http://www.math.binghamton.edu/dept/ComboSem/abstract.19990422coll.html

Neil Robertson The four-colour theorem Abstract for the Colloquium April 22, 1999 at 4:30

73. The Four-colour Theorem The fourcolour theorem Daniel P. Sanders , Yue Zhao, Coloring the faces ofconvex polyhedra so that like colors are far apart, Journal of Combinatorial http://portal.acm.org/citation.cfm?id=271675

74. LMS POSTER 5.15 6.00 Robin Thomas The four-colour theorem and beyond theRobertson-Seymour-Sanders-Thomas proof, recent progress on some generalisations, http://www.lms.ac.uk/meetings/poster.html

LONDON MATHEMATICAL SOCIETY BRITISH SOCIETY FOR THE HISTORY OF MATHEMATICS Joint Meeting THE FOUR-COLOUR PROBLEM Wednesday 23 October 2002 Cruciform Building, Lecture Theatre 1 University College London, Gower Street, London WC1 2.10 - 2.55 Robin Wilson: The four-colour problem: 1852-1940 [from Augustus De Morgan's letter to Henri Lebesgue's paper of 1940] 3.00 - 3.45 Kenneth Appel and Wolfgang Haken: Solving the four-colour problem [from Heesch’s contributions to the mathematical and computational aspects of the Appel-Haken solution] Tea (North Cloisters) LMS business 4.25 - 5.10 Dan Archdeacon: From the Heawood conjecture to topological graph theory [from the Ringel-Youngs solution of the Heawood conjecture to current developments in topological graph theory] 5.15 - 6.00 Robin Thomas: The four-colour theorem and beyond [the Robertson-Seymour-Sanders-Thomas proof, recent progress on some generalisations, and equivalent formulations inside and outside graph theory] A dinner will be held at The Old Amalfi Restaurant, 107 Southampton Row, London WC1 at 7.30 pm. The cost will be £24 per person, inclusive of wine, and a reception at De Morgan House beforehand. Those wishing to attend should inform Susan Oakes, Administrator, London Mathematical Society, enclosing a cheque payable to the ‘London Mathematical Society’ to arrive no later than Friday 18 October There are limited funds available to contribute in part to the expenses of members of the Society or research students to attend the meeting. Requests for support may be addressed to the Programme Secretary, London Mathematical Society, De Morgan House, 57-58 Russell Square, London WC1B 4HS (e-mail:

75. The Four Color Theorem Computer aided proof of the four color theorem by Neil Robertson, Daniel P.Sanders, Paul Seymour and Robin Thomas. http://www.math.gatech.edu/~thomas/FC/fourcolor.html

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 Table of Contents: History. Why a new proof? Outline of the proof. Main features of our proof. ... References. History. 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. He asked his brother Frederick if it was true that any map can be colored using four colors in such a way that adjacent regions (i.e. those sharing a common boundary segment, not just a point) receive different colors. Frederick Guthrie then communicated the conjecture to DeMorgan. The first printed reference is due to Cayley in 1878. A year later the first `proof' by Kempe appeared; its incorrectness was pointed out by Heawood 11 years later. Another failed proof is due to Tait in 1880; a gap in the argument was pointed out by Petersen in 1891. Both failed proofs did have some value, though. Kempe discovered what became known as Kempe chains, and Tait found an equivalent formulation of the Four Color Theorem in terms of 3-edge-coloring. The next major contribution came from Birkhoff whose work allowed Franklin in 1922 to prove that the four color conjecture is true for maps with at most 25 regions. It was also used by other mathematicians to make various forms of progress on the four color problem. We should specifically mention Heesch who developed the two main ingredients needed for the ultimate proof - reducibility and discharging. While the concept of reducibility was studied by other researchers as well, it appears 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.

76. Four-Color Theorem -- From MathWorld The fourcolor theorem states that any map in a plane can be colored using four-colors Cahit, I. Spiral Chains A New Proof Of The four Color theorem. http://mathworld.wolfram.com/Four-ColorTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Discrete Mathematics Graph Theory Graph Coloring ... Flawed Proofs Four-Color Theorem The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. This problem is sometimes also called Guthrie's problem after F. Guthrie, who first conjectured the theorem in 1853. The conjecture was then communicated to de Morgan and thence into the general community. In 1878, Cayley wrote the first paper on the conjecture. Fallacious proofs were given independently by Kempe (1879) and Tait (1880). Kempe's proof was accepted for a decade until Heawood showed an error using a map with 18 faces (although a map with nine faces suffices to show the fallacy). The Heawood conjecture provided a very general assertion for map coloring, showing that in a

77. Six-Color Theorem -- From MathWorld This number can easily be reduced to five, and the fourcolor theorem demonstratesthat the necessary number is, in fact, four. SEE ALSO four-Color theorem http://mathworld.wolfram.com/Six-ColorTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index DESTINATIONS About MathWorld About the Author Headline News ... Random Entry CONTACT Contribute an Entry Send a Message to the Team MATHWORLD - IN PRINT Order book from Amazon Discrete Mathematics Graph Theory Graph Coloring Six-Color Theorem To color any map on the sphere or the plane requires at most six-colors. This number can easily be reduced to five, and the four-color theorem demonstrates that the necessary number is, in fact, four. SEE ALSO: Four-Color Theorem Heawood Conjecture Map Coloring [Pages Linking Here] REFERENCES: Franklin, P. "A Six Colour Problem." J. Math. Phys. Hoffman, I. and Soifer, A. "Another Six-Coloring of the Plane." Disc. Math. Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, 1986. CITE THIS AS: Eric W. Weisstein. "Six-Color Theorem." From MathWorld A Wolfram Web Resource. http://mathworld.wolfram.com/Six-ColorTheorem.html Wolfram Research, Inc.

78. Ideas, Concepts, And Definitions four Color theorem. (See also The Mathematics Behind the Maps, The Most ColorfulMath of All, and The Story of the Young Map Colorer.) http://www.cs.uidaho.edu/~casey931/mega-math/gloss/math/4ct.html

Four Color Theorem (See also: The Mathematics Behind the Maps The Most Colorful Math of All , and The Story of the Young Map Colorer The Four Color Problem was famous and unsolved for many years. Has it been solved? What do you think? The story of the problem Since the time that mapmakers began making maps that show distinct regions (such as countries or states), it has been known among those in that trade, that if you plan well enough, you will never need more than four colors to color the maps that you make. The basic rule for coloring a map is that no two regions that share a boundary can be the same color. (The map would look ambiguous from a distance.) It is okay for two regions that only meet at a single point to be colored the same color, however. If you look at a some maps or an atlas, you can verify that this is how all familiar maps are colored. 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. When mathematicians picked up the thread of the conversation, they began by asking questions like: Are you sure that four colors are enough? How do you know that no one can draw a map that requires five colors? What is it about the way that regions are arranged and touch each other in a map that would make such a thing true? When the question came to the European mathematics community at the end of the 19th century, it was perceived as interesting but solvable. Prominent and experienced mathematicians who tackled the problem were surprised by their inability to solve it. Take for example, this account from

79. The Four Color Theorem The four Color theorem asserts that every planar graph and therefore every map In this context the four color theorem tells us that a space of four http://www.mathpages.com/home/kmath266/kmath266.htm

The Four Color Theorem How many different colors are sufficient to color the countries on a map in such a way that no two adjacent countries have the same color? The figure below shows a typical arrangement of colored regions. Notice that we define adjacent regions as those that share a common boundary of non-zero length. Regions which meet at a single point are not considered to be "adjacent". 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. We can just as well represent each country by a single point (vertex), and the adjacency between two bordering countries can be represented by a line (edge) connecting those two points. It's understood that connecting lines cannot cross each other. A drawing of this kind is called a planar graph. A simple map (with just five "countries") and the corresponding graph are shown below. A graph is said to be n-colorable if it's possible to assign one of n colors to each vertex in such a way that no two connected vertices have the same color. Obviously the above graph is not 3-colorable, but it is 4-colorable. The Four Color Theorem asserts that every planar graph - and therefore every "map" on the plane or sphere - no matter how large or complex, is 4-colorable. Despite the seeming simplicity of this proposition, it was only proven in 1976, and then only with the aid of computers. Notice that the above graph is "complete" in the sense that no more connections can be added (without crossing lines). The edges of a complete graph partition the graph plane into three-sided regions, i.e., every region (including the infinite exterior) is bounded by three edges of the graph. Every graph can be constructed by first constructing a complete graph and then deleting some connections (edges). Clearly the deletion of connections cannot cause an n-colorable graph to require any additional colors, so in order to prove the Four Color Theorem it would be sufficient to consider only complete graphs.

80. Four Color Theorem - Wikipedia, The Free Encyclopedia The four color theorem states that any plane separated into regions, The fourcolour theorem does not arise out of and has no origin in practical http://en.wikipedia.org/wiki/Four_color_theorem

Four color theorem From Wikipedia, the free encyclopedia. Example of a four color map The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. Each region must be contiguous - that is, it may not be partitioned as are Michigan and Azerbaijan 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. Ultimately, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.

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