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/

A NEW PROOF OF THE FOUR COLOUR THEOREM A NEW PROOF OF THE FOUR COLOUR THEOREM

2. Four Colour Theorem A Brief Historical Insight An essay on the history of the four color theorem by Dominic Verderaime. http://csci.adams.edu/~verderaimedj/FinalEssay/

3. The Four Color Theorem J. Math., 2 (1879), 193200. N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, The four colour theorem, J. Combin. Theory Ser. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

4. The Four Color Theorem 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

5. Four Colour Theorem THE four colour theorem http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

6. Four Color Theorem Intro Also see Nancy Casey Four Color Theorem and The MacTutor History of Mathematics archive - four colour theorem. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

7. Four Colour Theorem - Main The famous four colour theorem seems to have been first proposed by Möbius in 1840, four colour theorem. For any subdivision of the plane into http://www.geocities.com/dharwadker/main.html

Research Profile Teaching Software ... Get in touch... 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 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 at Drexel University for providing an elegant review of this website and its classification in both their Group Theory and Graph Theory categories; Thanks to the editors of at Akureyri University for writing an article about this proof; Thanks to Dr. Matrix for presenting this website with the Award for Science Excellence 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 III. EILENBERG MODULES IV. HALL MATCHINGS

8. The Four Colour Theorem The four colour theorem. Geometry and topology index History Topics Index. Version for printing http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

9. The Four Colour Theorem The four colour theorem was the first major theorem to be proved using a computer,having a proof that could not be verified directly by other 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 Version for printing 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.

10. Claim Of Proof To Four-Color Theorem Theorem 2. The four colour theorem Every standard map in a surface of unit connectivity can be expressed as a real construction of two http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

11. The Four Colour Theorem References DA Holton and S Purcell, The four colour theorema short history, Austral PC Kainen, Is the four color theorem true?, Geombinatorics 3 (2) (1993), 41-56 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/References/The_four_colou

References for: The four colour theorem Version for printing 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

12. Philos2.tex So the four colour theorem became the Four Colour Problem for the second time in history. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

13. Four Color Theorem for which programs and data supplements are available by FTP. A survey (in PostScript) of the paper "The FourColour Theorem" is there also. http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

14. New Scientist Breaking News - Computer Generates Verifiable 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

15. 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 http://tmsyn.wc.ask.com/r?t=an&s=hb&uid=24312681243126812&sid=343126

16. Four Color Theorem archive at Saint Andrews has a nice article on the four colour theorem.A summary of a new proof of the four color theorem is at Georgia Tech for which 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.

17. Sci.math FAQ: The Four Colour Theorem An equivalent combinatorial interpretation is Theorem 3 four colour theorem A recent simplification of the four colour theorem proof, by Robertson, 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 Related questions and answers Usenet FAQs Search Web FAQs ... RFC Index Send corrections/additions to the FAQ Maintainer: alopez-o@neumann.uwaterloo.ca Last Update June 15 2004 @ 00:31 AM

18. Four Colour Map Theorem From FOLDOC four colour theorem . four colour map theorem. mathematics, application (Or fourcolour theorem ) The theorem stating that if the plane is divided http://foldoc.doc.ic.ac.uk/foldoc/foldoc.cgi?four colour theorem

19. Four Colour Map Theorem From FOLDOC mathematics, application (Or four colour theorem ) The theorem stating thatif the plane is divided into connected regions which are to be coloured so http://foldoc.doc.ic.ac.uk/foldoc/foldoc.cgi?four colour map theorem

20. Claim Of Proof To Four-Color Theorem Theorem 2. The four colour theorem Every standard map in a surface of unitconnectivity can be expressed as a real construction of two sets of simple closed 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

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