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Check Out Help Search More Search Options Browse Pure Mathematics Pure Mathematics Never Miss an OUP Sale! Free Email Alerts! More Oxford email lists Enter Sales Promo Code Related Links: Table of Contents Higher Education Examination Copy Request Higher Education Comment Card Subjects ... Pure Mathematics Graphs, Colourings and the Four-Colour Theorem Robert A. Wilson paper 154 pages Also In Stock : hardback Mar 2002 In Stock Price: $85.00 $5.00 (US) $10.00 (INTL) Reviews Product Details About the Author(s) Learn more about this title... ... Graphs, Colourings and the Four-Colour Theorem Robert A. Wilson Price: $45.00 Browse More Titles In Pure Mathematics The Art of the Infinite Robert Kaplan An infinitely entertaining and informative journey into mathematical thinking Price: $26.00

63. ICHB Math Department - The Four Colour Problem It has taken more than 100 years before a correct proof for the fourcolour theorem has been found. The proof by Appel and Haken http://math.ichb.ro/modules.php?name=News&file=article&sid=82

64. Problem 2.1 Problem 2.1 fourColor theorem. N. Robertson, DP Sanders, P. Seymour,and R. Thomas The four-colour theorem , Manuscript. Abstract 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

65. Ideas, Concepts, And Definitions four Color theorem. (See also The Mathematics Behind the Maps, The MostColorful Math 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

66. The Four Color Theorem The four Color theorem. from Jodi Schneider I was interested in yourinfo about the fourColor theorem In 1976, the program written http://www.cs.uidaho.edu/~casey931/mmmail/4ct_mail.html

The Four Color Theorem from: Jodi Schneider I was interested in your info about the Four-Color Theorem... In 1976, the program written to prove this theorem took 1200 hours to run. Any idea what modern processing time would be? I'm curious if there might be a simpler solution (as math often develops from complex, incomplete solutions to simpler solutions) which mathematicians have not yet discovered. Thanks! -jodi Hi Jodi, I'm not sure how long the program would run now, or what type of machine it was run on back in 1976, only that it would still take extremely long. You are absolutely right about how a the first solution to a problem is complex and convoluted, but often it leads to a more simple one. Strangely, this has never been the case with the Four Color Problem, and there are many people who doubt that a less complex solution will ever be found. (A book I read recently said that the most significant open problem in mathematics was to find a simple solution to the Four Color Problem.) It has also been suggested that there are perhaps more categories of problems that we would ordinarily think. Instead of simply having problems that are solved and problems that are not solved, it could be that there are categroies that could resemble these: problems that can be solved simply, problems that can be solvedbut not simply, problems that are solvablebut not solved, and problems that are unsolvable. Take a look at the info on computational complexity , too. Nancy

67. Four-Color Theorem -- From MathWorld fourColor theorem. Robertson, N.; Sanders, D. P.; and Thomas, R. The four-Colortheorem. http//www.math.gatech.edu/~thomas/FC/fourcolor.html. http://mathworld.wolfram.com/Four-ColorTheorem.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook 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

68. The Four Color Theorem The four Color theorem. How many different colors are sufficient tocolor the countries on a map in such a way that no two adjacent 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.

69. Four Color Theorem - Wikipedia, The Free Encyclopedia four color theorem. Conceptually, a constraint such as this enables the mapto become nonplanar, and thus the four color theorem no longer applies. http://en.wikipedia.org/wiki/Four_color_theorem

Four color theorem From Wikipedia, the free encyclopedia. Example of a four colour map 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. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie . 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 have been done by Croatian mathematician Danilo Blanusa in the 40s.by finding original snark . However, it was not until 1977 that the four-color conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson Daniel Sanders Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone.

70. Four Color Theorem four color theorem. Robin Thomas, An Update on the fourColor theorem, Noticesof the American Mathematical Society, Volume 45, number 7 (August 1998). http://www.fact-index.com/f/fo/four_color_theorem.html

Main Page See live article Alphabetical index Four color theorem 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. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie. 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. However, it was not until 1977 that the conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. 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. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See

71. Four Color Theorem - Encyclopedia Article About Four Color Theorem. Free Access, encyclopedia article about four color theorem. four color theorem in Free onlineEnglish dictionary, thesaurus and encyclopedia. four color theorem. http://encyclopedia.thefreedictionary.com/Four color theorem

Dictionaries: General Computing Medical Legal Encyclopedia Four color theorem Word: Word Starts with Ends with Definition The four color theorem states that every possible geographical map can be colored with at most four colors For alternate meanings, see color (disambiguation). Color (American English) or colour (most other variants of English, including British English, New Zealand English and Australian English) is a sensation caused by light as it interacts with the eye, brain, and our experience. The perception of color is also greatly influenced by nearby colors in the visual scene. The term color is also used for the property of objects that gives rise to these sensations. Click the link for more information. 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. This theorem was conjectured in 1853 by Francis Guthrie. 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. However, it was not until 1977 that the four-color conjecture was finally proven by Kenneth Appel Kenneth Appel is a mathematician who, in 1976 with colleague Wolfgang Haken at the University of Illinois in Urbana, solved one of the most famous problems in mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any adjacent "countries" sharing the same color.

72. Four Color Theorem - Reference Library four color theorem. Robin Thomas, An Update on the fourColor theorem, Noticesof the American Mathematical Society, Volume 45, number 7 (August 1998). http://www.campusprogram.com/reference/en/wikipedia/f/fo/four_color_theorem.html

Reference Library: Encyclopedia Main Page See live article Alphabetical index Four color theorem 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. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie. 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. However, it was not until 1977 that the conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. 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. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See

73. Bipartite Graphs And The Four-Color Theorem Email rmabry@pilot.lsus.edu. Although the proof Kempe gave for the four ColorTheorem (4CT) in 1879 was flawed, the methods he devised were still useful. http://www.lsus.edu/sc/math/rmabry/bica/4color4web.htm

Bipartite Graphs and the Four-Color Theorem Rick Mabry LSU in Shreveport Shreveport, LA 71115-2399 E-mail: rmabry@pilot.lsus.edu Although the proof Kempe gave for the Four Color Theorem (4CT) in 1879 was flawed, the methods he devised were still useful. In 1890, Heawood used a modification of Kempe's methods to prove that planar graphs can be five-colored (see [ ] and [ ]). (All graphs here will be assumed to have no loops and no parallel edges.) In a 1968 paper ([ ]), Stephen Hedetniemi used modifications of Kempe's ideas to show, among other things, that every planar graph can be decomposed into the edge disjoint union of three bipartite graphs. My purpose in this note is to make the observation that a stronger result is now possible, namely Proposition A. Every planar graph can be decomposed into the edge disjoint union of two bipartite graphs. 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 [ So as much as I would prefer to prove Proposition A directly, I am able only to cheat the reader by proving instead the elementary

74. Mathematical - The Four Color Theorem And Computer Proofs The four Color theorem and Computer Proofs. R, ecently, the famous four color theorem was solved by an extensive computer proof. http://ebtx.com/math/math04.htm

The Four Color Theorem and Computer Proofs R ecently, the famous "four color theorem" was solved by an extensive computer proof. The fact that this theorem required an extensive proof in the first place gives evidence of how far research mathematics has descended since Gauss and Euler. It is now a pathetic embarrassment being followed closely by theoretical physics and preceeded by philosophy and 'modern' art. The fundamental knowledge now lacking in mathematics are answers to questions such as: What constitutes a "proof"? What is induction and deduction? And . . . most importantly . . . What does the term "self-evident" mean in mathematics? For those unfamiliar with the Four Color Theorem it is simply: What is the maximum number of different colors on a map (two dimensional, standard) that can be 'forced out' by the requirement that no two same color areas can touch at more than a point? Examples: Here is a 5-color map, but, obviously, I could make the black area blue and the yelow area green instead and reduce the number of colors required to 3 . After a little inspection (very little), one can see that if four mutually touching color areas are drawn, the only way to 'force out' the fifth would be to put in another color such that it touches the other four yet does not break any of the former connections.

75. Four Color Theorem Definition Meaning Information Explanation four color theorem definition, meaning and explanation and more about four colortheorem. FreeDefinition - Online Glossary and Encyclopedia, four color theorem. http://www.free-definition.com/Four-color-theorem.html

A B C D ... Contact Beta 0.71 powered by: akademie.de PHP PostgreSQL Google News about your search term Four color theorem The four color theorem states that every possible geographical map can be colored with at most four color s 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. This theorem was conjectured in 1853 by Francis Guthrie. 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. However, it was not until 1977 that the conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson, Daniel Sanders, Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. 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. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See

76. Online Encyclopedia - Four Color Theorem , Encyclopedia Entry for four color theorem.Dictionary Definition of four color theorem. imagefourColorMapEx......Encyclopedia http://www.yourencyclopedia.net/Four_color_theorem.html

Encyclopedia Entry for Four color theorem Dictionary Definition of Four color theorem 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. Two regions are called adjacent if they share a border segment, not just a point. This theorem was conjectured in 1853 by Francis Guthrie . 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. However, it was not until 1977 that the conjecture was finally proven by Kenneth Appel and Wolfgang Haken . They were assisted in some algorithmic work by J. Koch. The proof reduced the infinitude of possible maps to 1,936 configurations (later reduced to 1,476) which had to be checked one by one by computer. The work was independently double checked with different programs and computers. In 1996, Neil Robertson Daniel Sanders Paul Seymour and Robin Thomas produced a similar proof which required checking 633 special cases. This new proof also contains parts which require the use of a computer and are impractical for humans to check alone. 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. Ultimately, one had to have faith in the correctness of the compiler and hardware executing the program used for the proof. See

77. Four Color Theorem - Wikipedia, The Free Encyclopedia Back to Encyclopedia Main Page Printable Version of this Page Encyclopediahelp PhatNav s Encyclopedia A Wikipedia . four color theorem. http://www.phatnav.com/wiki/wiki.phtml?title=Four_color_theorem

78. Mudd Math Fun Facts: Four Color Theorem Easy level four Color theorem. Figure 1 Figure 1. Are 1890. The fourcolor theorem is true for maps on a plane or a sphere. The answer http://www.math.hmc.edu/funfacts/ffiles/10003.4.shtml

hosted by the Harvey Mudd College Math Department Francis Su Any Easy Medium Advanced Search Tips List All Fun Facts Fun Facts Home About Math Fun Facts ... Other Fun Facts Features 1553358 Fun Facts viewed since 20 July 1999. Francis Edward Su From the Fun Fact files, here is a Fun Fact at the Easy level: Four Color Theorem Figure 1 Are four colors always enough to color any map so that no two countries that share a border (in more than single points) have the same color? It is easy to show that you need at least four colors, because Figure 1 shows a map with four countries, each of which is touching the other. But is four sufficient for any map? Francis Guthrie made this conjecture in 1852, but it remained unproven until 1976, when Wolfgang Haken and Kenneth Appel showed that it was true! Also, quite interestingly, this proof required the assistance of a computer to check 1,936 different cases that every other case can be reduced to! To date no one knows a quick short proof of this theorem. Presentation Suggestions: Draw a few pictures to illustrate why the problem is difficult, and why (as some might ask) it is not valid to try to "check by example".

79. The Four Color Problem The four Color theorem. On a plane, or a sphere, no more than four colorsis necessary to color a map . Students and the four color theorem. http://bhs.broo.k12.wv.us/discrete/4Color.htm

The Four Color Theorem On a plane, or a sphere, no more than four colors is necessary to color a "map". Regions sharing a border must be colored differently. The History The origins of this problem are lost in antiquity, but early maps usually used more than four colors, and no mention is made of the number of colors required. One of the first statements of the problem occurred in 1852, when Francis Guthrie noticed the property while coloring a map of England. He mentioned the problem to his brother Frederick, who then mentioned it to his teacher, Augustus DeMorgan . DeMorgan then wrote about the problem to Sir William Hamilton , who was not interested in the problem. Part of Sir Hamilton's lack of interest may have been the misconception that the problem reduced to showing that it is impossible for five regions to share a common border. But, the problem is more involved than that. Later, in 1878, Arthur Cayley published an inquiry wondering if the question had been answered. This generated a general interest in the problem, leading to many attempts to prove the theorem, but without success. For many years, amateur and, some not so amateur, mathematicians as well as school children attempted to solve the problem. Mathematics journals regularly received, and discarded (probably unread) supposed "proofs" of the Four-Color Theorem. 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.

80. NASAexplores Maps 2002: Map Coloring And The Four Color Theorem The fourColor theorem. For 100 years, mathematicians attempted unsuccessfullyto solve what came to be known as the four-Color theorem. http://www.nasaexplores.com/extras/maps/four_color.html

Map Coloring: How Many Colors? The Four- Color Theorem A Mathematical Challenge A basic rule for coloring a map is that no two regions that share a boundary can be the same color; otherwise, the map will be hard to read. (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 standard maps are colored. no more than four colors . This belief came about as a matter of practical experience, no mapmaker having encountered a map that actually required five colors. (Of course, for artistic or other reasons, a mapmaker may choose to use more than four colors.) Unlike mapmakers, mathematicians want to have mathematical proof that a thing is true, so they asked questions. "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?" For 100 years, mathematicians attempted unsuccessfully to solve what came to be known as the

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