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 The Four Color Theorem. N. Robertson, DP Sanders, PD Seymour and R. Thomas,The four colour theorem, J. Combin. Theory Ser. B. 70 (1997), 244. 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 FAQ The four colour theorem. There are reader questions on this topic! Help others by sharing your knowledge math Subject sci.math FAQ The four colour theorem Date 17 Feb 2000 225204 Version 7.5 The four colour theorem Theorem 2 four colour theorem Every planar map http://www.faqs.org/faqs/sci-math-faq/fourcolour

Usenet FAQs Search Web FAQs Documents ... RFC Index sci.math FAQ: The Four Colour Theorem There are reader questions on this topic! Help others by sharing your knowledge From: alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math 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 Rate this FAQ N/A Worst Weak OK Good Great Current Top-Rated FAQs Are you an expert in this area? Share your knowledge and earn expert points by giving answers or rating people's questions and answers! This section of FAQS.ORG is not sanctioned in any way by FAQ authors or maintainers. Questions strongly related to this FAQ: give a four color planar map by konono (3/23/2004) How were the manual methods used to solve the four colour problem inferior to those used... by Paula (1/10/2004) Somewhere I read: If every 3-regular, plane graph had a circle, the adjoining countries of... by jost (7/1/2003) 4 colour theorem: - Is there any elementary proof for maps that are rectangles paved by...

4. 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

The four colour theorem Geometry and topology index History Topics Index The Four Colour Conjecture first seems to have been made by Francis Guthrie . He was a student at University College London where he studied under De Morgan . After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan . Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them. De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:- A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. ...... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did.... Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):- I am not likely to attempt your quaternion of colour very soon.

5. 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

References for The four colour theorem Books N L Biggs, E K Lloyd and R J Wilson, Graph Theory 1736-1936 (Oxford, 1986). Articles: N L Biggs, E K Lloyd and R J Wilson, C S Peirce and De Morgan on the four-colour conjecture, Historia Mathematica G D Birkhoff, Collected mathematical papers Vol. III : The four color problem, miscellaneous papers (New York 1968). L Arias C, A chronological account of the various theories developed for solving the four-color problem (Spanish), Rev. Integr. Temas Mat. D A Holton and S Purcell, The four colour theorem-a short history, Austral. Math. Soc. Gaz. P C Kainen, Is the four color theorem true?, Geombinatorics Proceedings of the Seminar on the History of Mathematics (Paris, 1982), 43-62. T L Saaty, The four-color problem : assaults and conquest (New York, 1977). J Wilson, New light on the origin of the four-color conjecture, Historia Mathematica Main index History Topics Index Biographies Index Famous curves index ... Societies, honours, etc JOC/EFR September 1996 The URL of this page is: http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/The_four_colour_theorem.html

6. The Four Colour Theorem The four colour theorem. Geometry and topology index. History Topics Index. The Four Colour Conjecture first seems to have been made by Francis Guthrie. He was a student at University College London http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.h

The four colour theorem Geometry and topology index History Topics Index The Four Colour Conjecture first seems to have been made by Francis Guthrie . He was a student at University College London where he studied under De Morgan . After graduating from London he studied law but by this time his brother Frederick Guthrie had become a student of De Morgan . Francis Guthrie showed his brother some results he had been trying to prove about the colouring of maps and asked Frederick to ask De Morgan about them. De Morgan was unable to give an answer but, on 23 October 1852, the same day he was asked the question, he wrote to Hamilton in Dublin. De Morgan wrote:- A student of mine asked me today to give him a reason for a fact which I did not know was a fact - and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted, but not more - the following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented. ...... If you retort with some very simple case which makes me out a stupid animal, I think I must do as the Sphynx did.... Hamilton replied on 26 October 1852 (showing the efficiency of both himself and the postal service):- I am not likely to attempt your quaternion of colour very soon.

7. Four Colour Map Theorem mathematics, application (Or "four colour theorem") The theorem stating that if the plane is divided into connected the world), it is never necessary to use more than four colours http://www.linuxguruz.org/foldoc/foldoc.php?four colour map theorem

8. Four Colour Map Theorem From FOLDOC four colour theorem four colour map theorem. mathematics, application (Or "four colour theorem") The theorem stating that if the plane is divided into connected never necessary to use http://www.instantweb.com/D/dictionary/foldoc.cgi?four colour theorem

9. 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 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 II. STEINER SYSTEMS

10. Four Color Theorem Four Color Theorem. Around book. The MacTutor History of Mathematics archiveat Saint Andrews has a nice article on the four colour theorem. http://grail.cba.csuohio.edu/~somos/4ct.html

Four Color Theorem Around 1998 Paul Kainen and I worked on an approach to the Four Color Theorem. He is a co-author of a book on this topic reprinted by Dover Publications. AUTHOR Saaty, Thomas L. TITLE The four-color problem : assaults and conquest / Thomas L. Saaty and Paul C. Kainen. PUBLISH INFO New York : Dover Publications, 1986. DESCRIPT'N vi, 217 p. : ill. ; 21 cm. NOTE Includes bibliographical references (p. 197-211) and index. SUBJECTS Four-color problem. LC NO QA612.19 .S2 1986. DEWEY NO 511/.5 19. OCLC # 12975758. ISBN 0486650928 (pbk.) : $6.00. AUTHOR Saaty, Thomas L. TITLE The four-color problem : assaults and conquest / Thomas L. Saaty and Paul C. Kainen. PUBLISHER New York : McGraw-Hill International Book Co., c1977. DESCRIPTION ix, 217 p. : ill. ; 25 cm. NOTES Bibliography: p. 197-211. Includes index. OCLC NO. 3186236. ISBN 0070543828 : $23.00. We take a pair of triangulations of a polygon and four color the vertices such that no two of the same color are connected by an edge of the triangulations. Polygon triangulations are easy to represent using data structures and the topological considerations of planarity are avoided. This turns the problem into a combinatorial one. The planarity reduces to circular order along the polygon and the non-crossing of diagonals. The history of this approach going back to Hassler Whitney and other references to this approach are in the book.

11. Four Colour Map Theorem From FOLDOC mathematics, application (Or "four colour theorem") The theorem stating that if the plane is divided into adjacent regions have the same colour (as when colouring countries on a http://www.instantweb.com/D/dictionary/foldoc.cgi?four colour map theorem

12. Four Colour Map Theorem - FOLDOC Definition four colour map theorem. ( Or "four colour theorem") The theorem stating that if the plane is divided adjacent regions have the same colour (as when colouring countries on a http://www.nightflight.com/foldoc-bin/foldoc.cgi?four colour map theorem

13. Four Color Theorem Intro Four Color Theorem Intro. Also see Nancy Casey Four Color Theorem and TheMacTutor History of Mathematics archive - four colour theorem. Addendum http://www.jimloy.com/geometry/4color.htm

Return to my Mathematics pages Go to my home page Four Color Theorem Intro We have all seen maps in which adjacent countries (or areas) are colored with different colors, so we can easily see the boundaries between them. Mathematicians asked, "Just how many colors are necessary?" They weren't trying to help out the map makers who occasionally bungle the job (I have seen several maps with mistakes in the coloring). The mathematicians found this an interesting, and diabolically difficult puzzle. Of course, the Four Color Theorem (previously called the Four Color Conjecture) was recently proven (by Wolfgang Haken and Kenneth Appel using a super computer at the University of Illinois, in 1976), showing that four colors is all you ever need, on a plane map. That proof is very long, and I will not show it. Instead, let's prove a "three color theorem:" Three color theorem - More than three colors are required for some map or maps. Proof: Look at the diagram, above left. Can't color it with just three colors, can you? That was a little informal. But, that is essentially the proof. We wanted to show that three colors were not enough for some map. All we have to do is show a map that requires four colors, and we have proved our conjecture. I could have given some reasoning why you can't color this map with three colors. But it should be fairly obvious.

14. Claim Of Proof To Four-Color Theorem The recent announcement by two American computer scientists that they have a proofof the four colour theorem, although they certainly have not published a http://www.lawsofform.org/gsb/nature.html

Claim of Proof to Four-Color Theorem By G. Spencer-Brown To the Editors of Nature , 4 Little Essex Street, London W C 2 England copies to other interested journals From G Spencer-Brown, 2 St Peters St, Castle Hill, Cambridge England 17 December 1976 Sirs The recent announcement by two American computer scientists that they have a proof of the four colour theorem, although they certainly have not published a proof, coupled with the fact that they are widely reported as saying they believe that no simple or elegant proof of this theorem is possible, prompts me to refer to the work of me and my brother, the late D J Spencer-Brown, on this theorem as early as 1960-1964. As reported in 1969 [ ], we found during this period an extremely elegant way of expressing the four-colour conjecture (as it then was) which, if verified, would lead to a correspondingly elegant proof. As is well known, the difficulty of the foul colour problem stems from the fact that the Heawood formulae [ ], say Hmin, Hmax, giving the minimum and maximum values for the chromatic numbers of surfaces (Sg) of connectivity g, give Hmin = Hmax = [(1/2)(7 + (24g - 23)^(1/2) )] for g > 1

15. LookSmart - Directory - Four-Color Theorem four colour theorem Find an overview of Francis Guthrie s Four ColourConjecture. Includes reference articles and related links. http://search.looksmart.com/p/browse/us1/us317914/us328800/us10163846/us10165260

@import url(/css/us/style.css); @import url(/css/us/searchResult1.css); Home IN the directory this category YOU ARE HERE Home Sciences Mathematics Graph Theory ... By Subject Four-Color Theorem - Offers discussion and analysis of the Four-Color Theorem in mathematics. Directory Listings About Bill Cherowitzo - Graph Theory Lecture Notes 5 Learn about the Four Color Theorem, Kuratowski's Theorem, and Koenig's Theorem. Brooke High School - Four Color Problem Read the history of the Four Color Theorem and its significance in graph theory. Four Color Theorem Find a proof that provides a solution to the famous problem about covering a map with four colors, which dates back to Francis Guthrie in 1852. Four Colour Theorem Find an overview of Francis Guthrie's Four Colour Conjecture. Includes reference articles and related links. Hamid V. Ansari - Solution to Four-Color Problem Read this mathematician's proof of the Four-Color Theorem. Written in 2002. Harvey Mudd College Fun Facts - Four Color Theorem Designated as "easy" in the fact collection, this page describes the Four Color Theorem in simple terms. Briefly mentions the colors needed for a torus. Jim Loy - Four Color Theorem Intro Written in layman's terms, this site informally "proves" the Four-Color Theorem and discusses other graph coloring issues.

16. LookSmart - Directory - Four-Color Theorem New Proof of the four colour theorem Ashay Dharwadker uses Eilenberg modules, Hallmatchings, Riemann surfaces, Steiner systems, and a map of Madhya Pradesh to http://search.looksmart.com/p/browse/us1/us317914/us328800/us10163846/us10165260

17. Four Colour Theorem - Encyclopedia Article About Four Colour Theorem. Free Acces encyclopedia article about four colour theorem. four colour theorem in Free onlineEnglish dictionary, thesaurus and encyclopedia. four colour theorem. http://encyclopedia.thefreedictionary.com/Four colour theorem

Dictionaries: General Computing Medical Legal Encyclopedia Four colour 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.

18. The Four Colour Theorem The four colour theorem. 21, 1977, pp. 429567. N. Robertson, D. Sanders,P. Seymour, R. Thomas The four colour theorem Preprint, February 1994. http://db.uwaterloo.ca/~alopez-o/math-faq/node56.html

Next: The Trisection of an Up: Famous Problems in Mathematics Previous: Famous Problems in Mathematics The Four Colour Theorem An equivalent combinatorial interpretation is 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. The number of basic forms, or configurations has been reduced to 633.

19. 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.

20. Four Colour Theorem four colour theorem. http://www.virtualology.com/virtualpubliclibrary/hallofeducation/Mathematics/Fou

You are in: Virtual Public Library Hall of Education Mathematics Four Colour Theorem Four Colour Theorem Start your search on Four Colour Theorem Other educational search engines: Ask Jeeves for Kids Britannica.com CyberSleuth Kids Education World ... Yahooligans Unauthorized Site: This site and its contents are not affiliated, connected, associated with or authorized by the individual, family, friends, or trademarked entities utilizing any part or the subject’s entire name. Any official or affiliated sites that are related to this subject will be hyper linked below upon submission and Virtualology's review. 2000 by Virtualology TM . All rights reserved. Virtualology TM Search: About Us e-mail us Now Available in Paperback President Who? Forgotten Founders Click Here Editing Sponsor Editing Sponsors review and select all student work for publishing. To learn more about our editing sponsor program click here. Published Students Primary Secondary College Graduate ... Technical

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