21. Sci.math FAQ: The Four Colour Theorem Subject sci.math FAQ The four colour theorem. 429567. N. Robertson, D. Sanders,P. Seymour, R. Thomas The four colour theorem Preprint, February 1994. http://www.cs.uu.nl/wais/html/na-dir/sci-math-faq/fourcolour.html

Note from archivist@cs.uu.nl : This page is part of a big collection of Usenet postings, archived here for your convenience. For matters concerning the content of this page , please contact its author(s); use the source , if all else fails. For matters concerning the archive as a whole, please refer to the archive description or contact the archivist. Subject: sci.math FAQ: The Four Colour Theorem This article was archived around: 17 Feb 2000 22:52:04 GMT All FAQs in Directory: sci-math-faq All FAQs posted in: sci.math Source: Usenet Version Archive-name: sci-math-faq/fourcolour Last-modified: February 20, 1998 Version: 7.5 http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick

22. 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://csai03.is.noda.sut.ac.jp/foldoc/foldoc.cgi?four colour theorem

23. 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://www.instantweb.com/D/dictionary/foldoc.cgi?four colour theorem

24. Sci.math FAQ: The Four Colour Theorem sci.math FAQ The four colour theorem. 21, 1977, pp. 429567. N. Robertson, D. Sanders,P. Seymour, R. Thomas The four colour theorem Preprint, February 1994. http://www.uni-giessen.de/faq/archiv/sci-math-faq.fourcolour/msg00000.html

Index sci.math FAQ: The Four Colour Theorem Subject : sci.math FAQ: The Four Colour Theorem From alopez-o@neumann.uwaterloo.ca (Alex Lopez-Ortiz) Date : 17 Feb 2000 22:52:04 GMT Newsgroups sci.math news.answers sci.answers Summary : Part 17 of 31, New version http://www.cs.unb.ca/~alopez-o Assistant Professor Faculty of Computer Science University of New Brunswick Index: Index sci-math-faq.fourcolour

25. Four Colour Theorem four colour theorem. Start your search on four colour theorem. Othereducational search engines Ask Jeeves for Kids Britannica 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 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

26. Four Colour Theorem Denis Howe dbh@doc.ic.ac.uk . Previous four colour map theorem NextFourier transform. four colour theorem. four colour map theorem. http://burks.brighton.ac.uk/burks/foldoc/90/44.htm

The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: four colour map theorem Next: Fourier transform four colour theorem four colour map theorem

27. Four Colour Map Theorem uk . Previous fourcolour glossies Next four colour theorem. fourcolour map theorem. mathematics, application (Or four colour http://burks.brighton.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

28. The Four Colour Theorem Question is there an actual equation for the four colour map theorem? i cannotfind it anywhere!! if so can you give a breif description on how it works!! http://mathcentral.uregina.ca/QQ/database/QQ.09.02/rhonda1.html

Name: rhonda Who is asking: Student Level: Secondary Question: is there an actual equation for the four colour map theorem? i cannot find it anywhere!! if so can you give a breif description on how it works!! Thank you Hi Rhonda, The theorem is just that, a theorem, and it proves that any map in the plane can be coloured properly with just 4 colours. That is not the same as saying that there is an efficient algorithm to 4-colour any planar map; it just assures us that it can actually be done. Thus unless your map is quite small and not too complicated it may be a very difficult task indeed to actually colour it properly with 4-colours. Cheers, Penny Go to Math Central To return to the previous page use your browser's back button.

29. Citations: A New Proof Of The Four Colour Theorem - Robertson, Sanders, Seymour, N. Robertson, DP Sanders, PD Seymour and R. Thomas, A new proof of the four colourtheorem, Electron. Res. A new proof of the fourcolour theorem. Electron. http://citeseer.nj.nec.com/context/1083701/372465

7 citations found. Retrieving documents... N. Robertson, D. P. Sanders, P. Seymour and R. Thomas, A new proof of the four-colour theorem , Electronic Research Announcements of the American Mathematical Society, vol. 2 (1996), 1725. Home/Search Document Details and Download Summary Related Articles ... Check This paper is cited in the following contexts: On Local Transformation of Polygons with Visibility.. - Hernando, Houle, Hurtado (Correct) ....the Platonic solids. In recent years, the use of computers has allowed the investigation of structures far more complex than those considered by the ancients. In graph theory, the only proofs yet known of the famous Four Color Theorem have been obtained by means of computer enumeration of cases Also, a general purpose program for computing automorphism groups of graphs and digraphs developed by Brendan McKay, Nauty, has been used to prove a number of graph theoretic results, most notably C. Hernando and F. Hurtado partially supported by CUR Gen. Cat . 1999SGR00356 and MEC DGES SEUID .... N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, `

30. Four-Coloring Six-Regular Graphs On The Torus - Collins, Hutchinson (ResearchInd colorable (context) Appel, Haken - 1976 87 Introduction to Graph Theory (context)- West - 1996 20 The four colour theorem (context) - Robertson, Sanders et http://citeseer.nj.nec.com/collins99fourcoloring.html

Four-Coloring Six-Regular Graphs On The Torus (1999) (Make Corrections) (1 citation) Karen L. Collins, Joan P. Hutchinson Home/Search Context Related View or download: wesleyan.edu/public foursixsent3.ps Cached: PS.gz PS PDF DjVu ... Help From: wesleyan.edu/vita (more) Homepages: K.Collins J.Hutchinson HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: this paper we focus on the torus and on ostensibly the most dicult case when the graph is a six-regular triangulation; in [28] Thomassen asks for a characterization of the non-4-colorable, 6-regular graphs on the torus. A theorem of Altshuler [4] shows that every 6regular toroidal graph can be represented as a 6-regular shifted rectangular grid on the torus; see Figure 2. A shift of one is the same as an unshifted grid. This allows us to focus rst on 6-regular grids and then to extend our... (Update) Context of citations to this paper: More ...can be 3 colored. This result is attributed to Heawood [16] though it could equally well be attributed to Kempe [20] As conjectured in , these graphs embedded on orientable surfaces with sufficiently large width can always be 4 colored [19] and so a 4 color extension Cited by: More Extending Colorings of Locally Planar Graphs - December Michael Albertson (Correct) Active bibliography (related documents): More All Long cycles in graphs on a fixed surface - Böhme, Mohar, al. (1999)

31. ColourTheory The four colour theorem (or the Four Color Theorem). Mathematicians Nomore so than in the area of the four colour theorem. This http://www.adit.co.uk/html/colourtheory.html

Maps and Mapping What is a Map Map Distortions How to build maps Using a GPS survey ... RGB and CYMK Colours The Four Colour Theorem (or the Four Color Theorem) Mathematicians and map makers share a lot of common ground. No more so than in the area of the four colour theorem. This theorem simply states that any map in a single plane can be coloured using four-colours in such a way that any regions sharing a common boundary (other than a single point) do not share the same colour. The theorem was first propounded by F Guthrie in 1853. Fallacious proofs have come and gone starting with Kempe in 1879 and Tait an 1880. In 1977 K. Appel and W. Haken used computer assistance to test many different combinations to effectively prove that four colours was all that was required in all instances. Since then, it may be that a mathematical proof has, at last, been arrived at. So, if we know that we can colour any map using just four colours how to we go about it. A little though would indicate that the problem is not straightforward. Simply starting with a random colour and an arbitrary polygon would soon lead to an impasse when the process met an area bounded by more than three other areas yet to be coloured. Kempe is credited with first recording that, when tackling a map of national boundaries, those with three or fewer neighbours presented no problems. His solution was to ignore (temporarily remove from the map) those countries with three or fewer neighbours. This process will immediately simplify the remainder of the map and can be repeated until only countries with three or fewer neighbours remain. The remaining areas can then be coloured. Then the missing countries can be restored in reverse order to their removal and coloured as the process proceeds. This effective procedure is know as Kempe transformations.

32. Map Colours Colour and Maps. The four colour theorem says that the areas of a mapcan be coloured in using just four colours. Ignore the minimalists http://www.adit.co.uk/html/map_colours.html

Maps and Mapping What is a Map Map Distortions How to build maps Using a GPS survey ... RGB and CYMK Colours Colour and Maps The four colour theorem says that the areas of a map can be coloured in using just four colours. Ignore the minimalists - colour has an important part to play in communicating information held on a map. Wise colour choice and colour usage can enhance the experience of your map users. The poor use of colour can degrade you map and make it difficult to use. Good colour choice like any design issue seems to demand a little flair and it is difficult to define hard a fast rules. Review other maps you have seen that have a similar purpose to your map. Try and decide where the use of colour has enhanced the objectives and where it has detracted from the map. This may help you decide upon how to apply colour to your specific map. If your map is likely to be photocopied or faxed (perhaps a “how to find us” map) then this should be taken into account when designing it. A monochrome line drawing might work better than a map with coloured areas and text. Reds and blacks fax well, blues and greens do not. If place and location names or any other form of text are an important attribute of your map then your colour scheme should take this into account. Colour contrasts could become very important to ensure that your text stands out on the relevant background areas.

33. Maths Thesaurus: Four Colour Theorem Home four colour theorem Any network of points and lines in a plane can be colouredin using no more than four colours, in such a way that no two adjacent http://thesaurus.maths.org/dictionary/map/word/3028

Four colour theorem Any network of points and lines in a plane can be coloured in using no more than four colours, in such a way that no two adjacent regions have the same colour. Find similar words More general: Compare with: Defined earlier: Theorem Seven colour theorem Four color problem New search: Age: 19 Choose Language and Age Make a suggestion nrich@damtp.cam.ac.uk

34. Count On emphasise the changes due to the morphing (A large version of the tiling is downloadableas an Acrobat PDF file as MT7.pdf) The four colour theorem A tiling http://www.mathsyear2000.org/explorer/morphing/13usedownload.shtml

Using the Morphing Tilings downloads There are a number of morphing tilings on the Downloads page . The following are a few suggestions for using them and any you might create yourself to explore patterns in new ways. Morphing patterns are more complex than simply repeating patterns and so offer more possibilities for creativity. Creating tilings can be a slow process if you draw them by hand, although a computer can speed up the process, with the right drawing package. Once you have a set of tiles then you can copy them to build up the tiling. To help you create tilings easily, you can also download a set of True Type fonts which allow you to create tilings in a word-processor. Type in a character and a tile appears. Click here for instructions on how to use these fonts. A starting point for designing your own tilings These pages plus the download files are meant as starting points from which to develop your own ideas. We hope you will find new ways to create new tilings, not just copy the ones here. Analysing the patterns in the tilings Everyone sees something different in morphing tilings. As the eye wanders over them it is possible to see the changes in different ways - sometimes you can focus on a local pattern and sometimes you see a movement over a larger area which is hard to pin down. They are ideal for teaching analytical skills and being able to put discoveries into words. Seeing what is happening is only the first step in showing someone else what you can see.

35. LinuxGuruz Foldoc Page Affordable Web Hosting. Cheap Web Hosting. Was our site helpfull? Want to make adonation? $. LinuxGuruz Foldoc. four colour theorem . four colour map theorem. http://foldoc.linuxguruz.org/foldoc.php?four colour theorem

36. 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

37. Four Color Theorem the book. The MacTutor History of Mathematics archive at Saint Andrewshas a nice article on the four colour theorem. A summary 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.

38. The Four Colour Theorem The four colour theorem. The Four Colour errors. The four colour theoremreturned to being the Four Colour Conjecture in 1890. Percy John http://math.5u.com/The four colour theorem.htm

Cheap Web Site Hosting The four colour theorem 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.

39. Bookmarks Four Colors QUESTIONS FAQ. sci.math FAQ The four colour theorem http//www.cs.ruu.nl/wais/html/na-dir/sci-math-faq/fourcolour.html up. http://perso.wanadoo.fr/patrick.davalan/Liens/liens_4colors.html

Pages Web Pages Personnelles Logiciels Papiers ... Faq Avertissement : PAGES WEB 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. The Geometry Junkyard Coloring Graph Coloring Page by Joseph Culberson. PAGES PERSONNELLES - HOME PAGES Robin Thomas School of Mathematics Georgia Institute of Technology Atlanta, Georgia Daniel P. Sanders LOGICIELS - SOFTWARES four at ftp.math.gatech.edu This directory contains the programs and data that serve as supplements to the paper `The Four-Colour Theorem' by N. Robertson, D. P. Sanders, P. D. Seymour and R. Thomas, J. Combin. Theory Ser. B 70 (1997), 2-44. They naturally fall into two parts, Reducibility and Discharging. Graph Coloring Programs Joseph Culberson Department of Computing Science University of Alberta Edmonton, Alberta, Canada T6G 2H1 Network Resources for Coloring a Graph DOCUMENTS - PAPERS four colors survey A new proof of the four-colour theorem Neil Robertson, Daniel P. Sanders, Paul Seymour, and Robin Thomas couleurs Bienvenue chez Sylvain Poirier JOURNAUX - LETTERS Electronic Research Announcements of the American Mathematical Society A new proof of the four-colour theorem QUESTIONS - FAQ sci.math FAQ: The Four Colour Theorem

40. University Of Wuppertal - Dep. Of Mathematics - Poster imaginary maps. It has taken more than 100 years before a correctproof for the four colour theorem has been found. The proof by http://www.math.uni-wuppertal.de/org/Poster/TextE.htm

The four colour problem Primes Crash Test Computerized Tomography The four colour problem In 1852, while colouring a map representing the english counties, the british mathematician Francis Guthrie realized that only four colours where necessary to satisfy the criterion that neighbouring counties should have different colours. It turns out that this is true for any (real or imaginary) map. The poster shows such a colouring for the countries of Europe and, in small, for three other imaginary maps. It has taken more than 100 years before a correct proof for the four colour theorem has been found. The proof by Appel and Haken in 1976 has solicited much of a controversial discussion since it heavily relies on computer calculations. These are so extensive that humans cannot verify them "by hand". Consequently, proving the four colour theorem correct also means proving that the program is implemented correctly and that the computer works correctly. Instead of taking Europe and its countries, we could illustrate the four colour theorem with Germany and its regions. This can be found at MathePrisma (in German). There you can also colour some other maps on your own and develop strategies to get the right colouring fast.

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