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How To Almost Prove The 4-color Theorem « Secret Blogging Seminar Oct 7, 2009 First I ll explain a standard reduction of the 4color theorem to a question about 3-coloring edges of trivalent graphs. http://sbseminar.wordpress.com/2009/10/07/how-to-almost-prove-the-4-color-theore
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Extractions: From Wikipedia, the free encyclopedia Jump to: navigation search Example of a four color map The four color theorem is a theorem of mathematics . It says that in any plane surface with regions in it (people think of them as maps ), the regions can be colored with no more than four colors . Two regions that have a common border must not get the same color. They are called adjacent (next to each other) if they share a segment of the border, not just a point. This was the first theorem to be proved by a computer , in a proof by exhaustion . In proof by exhaustion, the conclusion is established by dividing it into cases, and proving each one separately. The number of cases sometimes may be very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because most of the cases were checked by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases. It is quite obvious that there are maps that cannot be colored in this way with only three colors. On the other hand, four colors are enough for every map.
A New Proof Of The Four-Colour Theorem This journal is archived by the American Mathematical Society. The master copy is available at http//www.ams.org/era/ http://www.mpim-bonn.mpg.de/era-mirror/1996-01-003/1996-01-003.html
Extractions: @import "/css/gridmain.css"; @import "/css/article.css"; @import "/css/comlist.css"; @import "/data/images/ns/haas/haas.css"; SUBSCRIBE TO NEW SCIENTIST Select a country United Kingdom USA Canada Australia New Zealand Other Log in Email Password Remember me Your login is case sensitive A computer-assisted proof of a 150-year-old mathematical conjecture can at last be checked by human mathematicians. The Four Colour Theorem, proposed by Francis Guthrie in 1852, states that any four colours are the minimum needed to fill in a flat map without any two regions of the same colour touching. A proof of the theorem was announced by two US mathematicians, Kenneth Appel and Wolfgang Haken, in 1976. But a crucial portion of their work involved checking many thousands of maps - a task that can only feasibly be done using a computer. So a long-standing concern has been that some hidden flaw in the computer code they used might undermine the overall logic of the proof. But now Georges Gonthier, at Microsoft's research laboratory in Cambridge, UK, and Benjamin Werner at INRIA in France have proven the theorem in a way that should remove such concerns.
An Update On The Four-Color Theorem An Update on the FourColor Theorem Robin Thomas 848 N OTICES OF THE AMS V OLUME 45, N UMBER 7 E very planar map of connected countries can be colored using four colors in such a way that http://www.ams.org/notices/199807/thomas.pdf
Extractions: Skip to Navigation Search this site: Issue 10 Submitted by plusadmin on January 1, 2000 in January 2000 In this final article in our series on Proof, we examine the philosophy of mathematical proof. What precisely is a proof? The answer seems obvious: starting from some axioms , a proof is a series of logical deductions , reaching the desired conclusion. Every step in a proof can be checked for correctness by examining it to ensure that it is logically sound, and you can tell that you've proved a theorem once and for all by making sure that every step is correct. This might sound simple enough, but one problem is that humans (and even computers) are fallible: what if the person checking a proof for correctness makes a mistake and thinks that a step which is logically incorrect is in fact correct? Obviously somebody else will need to check that the person doing the checking didn't make any mistakes; and somebody will need to check that person, and so on. Eventually you run out of people who could check the proof: and in theory they could
Four Color Theorem 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 http://www.fact-index.com/f/fo/four_color_theorem.html
Extractions: Main Page See live article Alphabetical index 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
Algebraic Proof Of 4-Colour Theorem? - MathOverflow 4Colour Theorem. Every planar graph is 4-colourable. This theorem of course . Does Lie Algebras and the Four Color Theorem by Dror Bar-Natan qualify ? http://mathoverflow.net/questions/19240/algebraic-proof-of-4-colour-theorem
Extractions: login faq how to ask meta ... 4-Colour Theorem. Every planar graph is 4-colourable. This theorem of course has a well-known history. It was first proven by Appel and Haken in 1976, but their proof was met with skepticism because it heavily relied on the use of computers. The situation was partially remedied 20 years later, when Robertson, Sanders, Seymour, and Thomas published a new proof of the theorem. This new proof still relied on computer analysis, but to such a lower extent that their proof was actually verifiable. Finally, in 2005, Gonthier and Werner used the Coq proof assistant to formalize a proof, so I suppose only the most die hard skeptics remain. My question stems from reading this paper by Robin Thomas. In it, he describes several interesting reformulations of the 4-colour theorem. Here is one: bracketing Theorem. The surprising fact is that this innocent looking theorem implies the 4-colour theorem. Question.
4-colour Theorem For the latest word see Robin Thomas, An Update on the FourColor Theorem, Notices of the Amer. Math. Soc. 457 (August 1998) 848-859 (an authoritative http://mathcentral.uregina.ca/RR/database/RR.09.97/fisher1.html
Extractions: 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.
About "The Four Colour Theorem" Linked essay describing work on the theorem from its posing in 1852The Math Forum Internet Mathematics Library. The Four Colour Theorem http://mathforum.org/library/view/5799.html
Extractions: Visit this site: http://www-history.mcs.st-and.ac.uk/history/HistTopics/The_four_colour_theorem.html Author: MacTutor Math History Archives Description: Linked essay describing work on the theorem from its posing in 1852 through its solution in 1976, with two other web sites and 9 references (books/articles). Levels: High School (9-12) College Languages: English Resource Types: Articles Bibliographies Math Topics: Graph Theory History and Biography
Siwei Zhu Siwei Zhu Ph.D. student UCLA Department of Maths Office MS 3970 Email UCLA Maths Some Things Four colour theorem Polynomials are refinable Guide to Being a Student of Mathematics http://www.math.ucla.edu/~siwei/
THREE PROOFS FOR THE FOUR COLOR THEOREM BY SPIRAL-CHAINS The famous four color problem is to color the regions of a given map using least number of colors so that adjacent regions have different colors. http://www.emu.edu.tr/~cahit/the four color theorem --- three proofs.htm
Extractions: THE FOUR COLOR THEOREM AND THREE PROOFS I. Cahit Abstract In August 2004 the author has introduced spiral chains for the maximal planar graphs in the sequel of a non-computer proof of the famous four color theorem. A year later he has by the use of we have abled to extend the given incomplete four coloring of the bad-example to an proper four coloring of the whole maximal planar graph. Here the proof proposed uses double spiral chains which is started from the undecided node of the graph and traverses clock or counter-clockwise direction the other nodes in the shape of two parallel spirals (double-spirals). INTRODUCTION The famous four color problem is to color the regions of a given map using least number of colors so that adjacent regions have different colors. The story of the problem as well as its known proofs is very long and tedious [1],[6],[2]. There are several historical false elegant proofs [7],[9],[10-12]. Many books and articles have been published on the problem so I leave it to the reader to choose his own selection. Robin Thomas has listed many equivalent formulations of the four color theorem in [18]. My selection about the four color problem and theorem
Extractions: These pages are not updated anymore. They reflect the state of 20 August 2005 . For the current production of this journal, please refer to http://www.math.psu.edu/era/ This journal is archived by the American Mathematical Society. The master copy is available at http://www.ams.org/era/ Abstract. Retrieve entire article TeX source
Maps And The Four-Color Theorem File Format PDF/Adobe Acrobat Quick View http://www.ccmr.cornell.edu/education/modules/documents/MapsandtheFour-ColorTheo
Mathwire.com | June 2006 Four Color Theorem In 1976, mathematicians used a supercomputer to help prove that any map could be colored with four colors at most. http://mathwire.com/archives/june06.html
Extractions: Browse Math Topics Mathwire.com June 2006 Introduce elementary students to the discrete math concept of map coloring. The basic concept is coloring a map using the fewest colors possible with these two rules: Students may use a blank map of the United States or a state map of counties to experiment with map coloring skills. It's often helpful for students to use pencil to code in colors before actually coloring so that they can easily rearrange colors, as necessary. The challenge is to color the map using the fewest number of colors. Can the USA map be colored using only 2 colors, or do you need 3 or 4 or 5 colors? Allow students to work on the task and consult classmates to compare notes and discuss the process they are using to minimize colors. Four Color Theorem : In 1976, mathematicians used a super-computer to help prove that any map could be colored with four colors at most. Knowledge of this mathematical theory helps students work to color any map in 2-4 colors. Later, they will apply these same principles to graph theory.
Play Flood Fill, A Free Online Game On Kongregate Oct 27, 2009 Every map you draw like this can be coloured in the required way with 4 or less colours. Wikipedia four colour theorem if you dont believe http://www.kongregate.com/games/Onefifth/flood-fill
