1. Four Colour Theorem A new proof of the four color theorem by Ashay Dharwadker that uses group theory and Steiner systems. http://www.geocities.com/dharwadker/

2. The Four Color Theorem A brief summary of a new proof of the four color theorem by Neil Robertson, Daniel P. Sanders, Paul Category Science Math Combinatorics Graph Theory...... J. Math., 2 (1879), 193200. N. Robertson, DP Sanders, PD Seymour and R. Thomas,The four colour theorem, J. Combin. Theory Ser. B. 70 (1997), 2-44. 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.

3. Sci.math FAQ: The Four Colour Theorem sci.math news.answers sci.answers Subject sci.math FAQ The four colour theorem FollowupTo sci.math Date 17 Feb 2000 http://www.faqs.org/faqs/sci-math-faq/fourcolour

sci.math FAQ: The Four Colour Theorem From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math news.answers sci.answers Subject: sci.math FAQ: The Four Colour Theorem Followup-To: sci.math 88hu2k$qu6$1@watserv3.uwaterloo.ca alopez-o@neumann.uwaterloo.ca alopez-o@unb.ca ... http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick By Archive-name By Author By Category By Newsgroup ... Help Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update March 05 2003 @ 01:20 AM

4. Four Colour Theorem four colour theorem four colour theorem The four colour theorem states that any planar (flat 2d) map needs only 4 colours to be coloured in. There is only one rule, adjacent regions must be in different colours. http://www.users.globalnet.co.uk/~perry/maths/fourcolour/fourcolour.htm

Four Colour Theorem [Home] [Other Maths] Four Colour Theorem The four colour theorem states that any planar (flat 2d) map needs only 4 colours to be coloured in. There is only one rule, adjacent regions must be in different colours. Regions that are connected at a point are not adjacent. The above image shows a skeleton map that needs colouring, and two possible colourings. There are generally many alternatives using only 4 colours, and the Four Colour theorem asks 'Is there a map that we can create that requires 5 colours?'. The first colouring is created using an agressive technique - a colour is applied to a random region, and then, again randomly, to as many regions as possible. Once exhausted, another colour is used. The second counts the regions - 12 in this case, and then creates 3 regions of each colour. This raises an important point. We can colour the maps using 5 colours, but we only NEED four. And is it possible to create a scenario where 4 colours does not seem to be enough, but there will be an different pattern that does colour the map in four. A simple proof of this theorem has eluded mathematicians for centuries, but here is a proof that the four colour theorem is true, and that there is no planar map that needs 5 or more colours.

5. Four Colour Problem Four Color Theorem. The four colour theorem. Instead of reiterating the theoretical work on this subject, I have http://pc-han.dto.tudelft.nl/fcp

Four Colour Problem Main references on the Internet are: Four Color Theorem The four colour theorem Instead of re-iterating the theoretical work on this subject, I have succeeded in devising a rather general program , which attempts to do Four Colouring on Bit MaPs (BMP files) in practice. The final algorithm is optimal , in the sense that it uses the minimum number of colors . Most of the time 4 of them are required (according to the Four Color Theorem); see below at ' Best Results so far '. But there are several other goodies in the program too (and I find them equally important, at least): An algorithm which generates closed and oriented Contour lines, for quite arbitrary Black and White Bitmaps An algorithm for classifying all of the Countries, especially those with Enclaves inside (also with a robust definition of "inside") Determining the Neighbours of a Country, treated like solving a Radiative Heat Transfer problem in 2-D, where white pixels are the "opaque" medium Euclid's GCD (Greatest Common Divisor) Algorithm, for obtaining a discrete approximation to straight lines-of-sight through the B/W medium Executables Country Maps USA Africa BoxVille BumpTown ... Wheel shaped region, odd # countries

6. The Four Colour Theorem The four colour theorem. basic errors. The four colour theorem returnedto being the Four Colour Conjecture in 1890. Percy John Heawood http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html

7. The Four Colour Theorem References References for The four colour theorem. Books DA Holton and S Purcell, The fourcolour theorema short history, Austral. Math. Soc. Gaz. 6 (1) (1979), 11-14. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/References/The_four_colour_t

8. Four Colour Theorem Find an overview of Francis Guthrie's Four Colour Conjecture. Includes reference articles and related links. The four colour theorem. Geometry and topology index http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.h

9. About "A New Proof Of The Four Colour Theorem" A New Proof of The four colour theorem An article presenting a proof of the four color theorem that uses group theory and Steiner systems, illustrated using a map of Madhya Pradesh and adjoining states in India. http://mathforum.com/library/view/16622.html

A New Proof of The Four Colour Theorem Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.geocities.com/dharwadker/ Author: Ashay Dharwadker Description: An article presenting a proof of the four color theorem that uses group theory and Steiner systems, illustrated using a map of Madhya Pradesh and adjoining states in India. Introduction; Map Colouring; Steiner Systems; Eilenberg Modules; Hall Matchings; Riemann Surfaces; Main Construction; References. Levels: Research Languages: English Resource Types: Articles Math Topics: Group Theory Graph Theory Suggestion Box Home ... Search http://mathforum.org/ webmaster@mathforum.org

10. Four Colour Theorem - Main A NEW PROOF OF. THE four colour theorem. BY. ASHAY DHARWADKER. H501 PALAMVIHAR DISTRICT GURGAON HARYANA 1 2 2 0 1 7 INDIA four colour theorem. http://www.geocities.com/dharwadker/main.html

A NEW PROOF OF THE FOUR COLOUR THEOREM BY ASHAY DHARWADKER H-501 PALAM VIHAR DISTRICT GURGAON HARYANA 1 2 2 1 7 INDIA email dharwadker@yahoo.com Research Profile Teaching ... Software ACKNOWLEDGEMENTS Thanks to the Canadian Mathematical Society for selecting this website as a "cool math site of the week" and knot No. 221 in their popular braid of links on October 5, 2000; Thanks to the editors of the The Math Forum Internet Mathematics Library for providing a concise and elegant review of this website and its classification in both their Group Theory and Graph Theory categories; Thanks to the editors of for writing an article about this proof for Icelandic readers; Thanks to the editors of Look Smart for featuring this website in their Topology section; Thanks to Dr. Matrix for honouring this website with the Award for Science Excellence on May 14, 2002 and selecting it for prominent display in the categories of Mathematics and Creative Minds ; Thanks to the editors of Yahoo! for featuring this website in their list of Famous Mathematics Problems CONTENTS INTRODUCTION I. MAP COLOURING

11. About "A New Proof Of The Four Colour Theorem" From the Math Forum Internet Mathematics Library.Category Science Math Combinatorics Graph Theory References......A New Proof of The four colour theorem. Library Home Full Table ofContents Suggest a Link Library Help Visit this site http http://mathforum.org/library/view/16622.html

A New Proof of The Four Colour Theorem Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.geocities.com/dharwadker/ Author: Ashay Dharwadker Description: An article presenting a proof of the four color theorem that uses group theory and Steiner systems, illustrated using a map of Madhya Pradesh and adjoining states in India. Introduction; Map Colouring; Steiner Systems; Eilenberg Modules; Hall Matchings; Riemann Surfaces; Main Construction; References. Levels: Research Languages: English Resource Types: Articles Math Topics: Group Theory Graph Theory Suggestion Box Home ... Search http://mathforum.org/ webmaster@mathforum.org

12. Four Colour Map Theorem mathematics, application (Or "four colour theorem") The theorem stating that if the plane is divided into connected http://burks.bton.ac.uk/burks/foldoc/89/44.htm

The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: four-colour glossies Next: four colour theorem four colour map theorem mathematics application The proof, due to Appel and Haken, attained notoriety by using a computer to check tens of thousands of cases and is thus not humanly checkable, even in principle. Some thought that this brought the philosophical status of the proof into doubt. There are now rumours of a simpler proof, not requiring the use of a computer. See also chromatic number

13. About "The Four Colour Theorem" The four colour theorem. Library Home Full Table of Contents Suggest a Link Library Help Visit this site http//wwwhistory http://mathforum.org/library/view/5799.html

The Four Colour Theorem Library Home Full Table of Contents Suggest a Link Library Help 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 Suggestion Box Home ... Search http://mathforum.org/ webmaster@mathforum.org

14. Www.faqs.org/ftp/faqs/sci-math-faq/fourcolour From alopezo@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups sci.math,news.answers,sci.answersSubject sci.math FAQ The four colour theorem Followup-To http://www.faqs.org/ftp/faqs/sci-math-faq/fourcolour

Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!news-peer.gip.net!news.gsl.net!gip.net!news.maxwell.syr.edu!sunqbc.risq.qc.ca!News.Dal.Ca!torn!watserv3.uwaterloo.ca!undergrad.math.uwaterloo.ca!neumann.uwaterloo.ca!alopez-o From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math,news.answers,sci.answers Subject: sci.math FAQ: The Four Colour Theorem Followup-To: sci.math Date: 17 Feb 2000 22:52:04 GMT Organization: University of Waterloo Lines: 65 Approved: news-answers-request@MIT.Edu Expires: Sun, 1 Mar 1998 09:55:55 Message-ID:

15. Articles very diverse group of students. The Journey of the four colour theoremThrough Time. This article, written by Andreea S. Calude, is http://matholymp.com/ARTICLES/articles.html

Articles You need Acrobat Reader to read these articles or download them. Down With Zeros at IMO! This article, written by Derek Holton and Arkadii Slinko, summarizes the experience of USSR Mathematics Olympiad to cater for a very diverse group of students. The Journey of the Four Colour Theorem Through Time This article, written by Andreea S. Calude, is a historical survey of the Four Colour Theorem and a discussion of the philosophical implications of its computer-aided proof. Mathematics and Democracy This article proves a simplified form of famous Plott's and McKelvey's theorems about the dynamics that a competition of two political parties might have. You will learn how it is possible that two political parties can defeat each other in a series of elections moving further and further from political ideals of voters. You need to know only Pythagoras' theorem to read this article. Lots of fun. Some interesting reading on this topic: Accurate Democracy - see the best voting methods for elections and meetings; download PoliticalSim for learning and free software for voting. How to win a high school election.

16. Four Colour Map Theorem From FOLDOC four colour theorem . four colour map theorem. mathematics, application (Or four colour theorem ) The theorem stating that if http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?four colour theorem

17. The Four Colour Theorem The four colour theorem. Theorem 3 four colour theorem Every loopless planargraph admits a vertexcolouring with at most four different colours. http://db.uwaterloo.ca/~alopez-o/math-faq/mathtext/node27.html

Next: The Trisection of an Up: Famous Problems in Mathematics Previous: Famous Problems in Mathematics The Four Colour Theorem Theorem 2 [Four Colour Theorem] Every planar map with regions of simple borders can be coloured with 4 colours in such a way that no two regions sharing a non-zero length border have the same colour. An equivalent combinatorial interpretation is Theorem 3 [Four Colour Theorem] Every loopless planar graph admits a vertex-colouring with at most four different colours. This theorem was proved with the aid of a computer in 1976. The proof shows that if aprox. 1,936 basic forms of maps can be coloured with four colours, then any given map can be coloured with four colours. A computer program coloured these basic forms. So far nobody has been able to prove it without using a computer. In principle it is possible to emulate the computer proof by hand computations. The known proofs work by way of contradiction. The basic thrust of the proof is to assume that there are counterexamples, thus there must be minimal counterexamples in the sense that any subset of the graphic is four colourable. Then it is shown that all such minimal counterexamples must contain a subgraph from a set basic configurations. But it turns out that any one of those basic counterexamples can be replaced by something smaller, while preserving the need for five colours, thus contradicting minimality.

18. Famous Problems In Mathematics next up previous contents Next The four colour theorem Up Frequently Asked Questionsin Previous Names of Large Numbers. Famous Problems in Mathematics. http://db.uwaterloo.ca/~alopez-o/math-faq/node55.html

Next: The Four Colour Theorem Up: Frequently Asked Questions in Previous: Names of Large Numbers Famous Problems in Mathematics The Four Colour Theorem The Trisection of an Angle Which are the 23 Hilbert Problems? Unsolved Problems ... Twin primes conjecture Alex Lopez-Ortiz Mon Feb 23 16:26:48 EST 1998

19. The Four Colour Theorem The four colour theorem. Theorem 1 3 . The four colour theorem is thusequivalent to the following statement about graphs Theorem http://www.shef.ac.uk/~puremath/theorems/fourcol.html

The four colour theorem Theorem 1 Given a map drawn in the plane we can colour the countries with red, green, blue and orange in such a way that any two adjacent countries have different colours. Here countries are assumed to be connected; we disallow the United States, for example, as Alaska and Hawaii are separated from the main body of the country. Two countries are only considered to be adjacent if they have a common boundary of nonzero length, not if they just touch at a single point. The theorem is illustrated by the following colouring of the map of Africa: For a detailed history of this problem, see the St Andrews history of mathematics site. Briefly, Francis Guthrie conjectured in 1852 that the result was true, but was unable to prove it. Over the next 27 years a number of powerful mathematicians such as Cayley considered the question but were also unable to answer it. A proof was announced in 1879 by Alfred Kempe to great acclaim, but eleven years later Percy Heawood revealed a fatal error in his argument. Over the next 86 years steady progress was made, showing for example that every map with at most 95 countries can be coloured with four colours. In 1976, Appel and Haken gave a proof of the theorem, which required consideration of 1476 different special cases, with computer assistance. The four-colour theorem is thus unusual in at least two ways: one of very few major results for which an incorrect proof has remained undetected for a considerable time, and the first major result to be proved by computer. In 1996 Robertson, Sanders, Seymour and Thomas gave a

20. All Theorems Of The Week theorem of algebra; Gödel's Theorem; The four colour theorem; Thearithmeticgeometric mean inequality; Cantor's Diagonal Argument; http://www.shef.ac.uk/~puremath/theorems/all.html

Other Theorems of the Month Pythagorean triples and the congruent number problem Lagrange's four-square theorem Bezout's theorem The solution of cubics and quartics ... The School of Mathematics and Statistics

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