Five Color Theorem - Wikipedia, The Free Encyclopedia It is implied by the stronger four color theorem, but can be proved with considerablyless effort. edit. Outline of the proof http://en.wikipedia.org/wiki/Five_color_theorem
Four Color Theorem Intro Of course, the four Color theorem (previously called the four Color And thefour Color theorem becomes part of mathematical subject called Graph Theory. http://www.jimloy.com/geometry/4color.htm
Extractions: Go to my home page 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.
Four Color Theorem: Information From Answers.com four color theorem This article needs to be cleaned up to conform to a higherstandard of quality. See How to Edit and Style and Howto for help, or. http://www.answers.com/topic/four-color-theorem
Extractions: showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping four color theorem Wikipedia four color theorem This article needs to be cleaned up to conform to a higher standard of quality. See How to Edit and Style and How-to for help, or this article's talk page Example of a four color map The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than 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. Each region must be contiguous - that is it may not be partitioned as are Michigan and Azerbaijan It is obvious that three colors are inadequate, and it is not at all difficult to prove that five colors are sufficient to color a map. The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand. Ultimately, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof. The conjecture was first proposed in when Francis Guthrie , while trying to color the map of counties of England , noticed that only four different colors were needed. At the time, Guthrie was a student of
Extractions: Example of a four color map 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 colour. Two regions are called adjacent if they share a border segment, not just a point. The problem itself dates to 1852, when Francis Guthrie , while trying to color the map of counties of England, noticed that four colors sufficed. According to Augustus De Morgan A student of mine [Guthrie] 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... The first printed reference is due to Arthur Cayley in 1878, "On the colourings of maps.", Proc. Royal Geog. Soc. 1 (1879), 259-261.
Extractions: See How to Edit and Style and How-to for help, or this article's talk page . Remove this message when done. Example of a four color map The four color theorem states that every possible geographical map can be colored using no more than 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. It is obvious that three colors are inadequate, and it is not difficult, relatively speaking, to prove that five colors are sufficient to color a map. The four color theorem was the first major theorem to be proven using a computer, and the proof is not accepted by all mathematicians because it would be unfeasible for a human to verify by hand. Ultimately, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof. Contents 1 History
4-color Theorem | Lambda The Ultimate fourcolor theorem. Unfortunately, we do not have the refereeing resources tohave all such papers assessed properly. In view of this, we cannot offer you http://lambda-the-ultimate.org/node/view/864
Rick Mabry Rick Mabry, Bipartite Graphs and the four Color theorem, Although the proofKempe gave for the four Color theorem (4CT) in 1879 was flawed, http://www.lsus.edu/sc/math/rmabry/bica/4color4web.htm
Extractions: Although the proof Kempe gave for the Four Color Theorem (4CT) in 1879 was flawed, the methods he devised were still useful. In 1890, Heawood used a modification of Kempe's methods to prove that planar graphs can be five-colored (see [ ] and [ ]). (All graphs here will be assumed to have no loops and no parallel edges.) In a 1968 paper ([ ]), Stephen Hedetniemi used modifications of Kempe's ideas to show, among other things, that every planar graph can be decomposed into the edge disjoint union of three bipartite graphs. My purpose in this note is to make the observation that a stronger result is now possible, namely Proposition A. Every planar graph can be decomposed into the edge disjoint union of two bipartite graphs. It is not surprising that Hedetniemi did not obtain the result of Proposition A, since
Extractions: The Four color theorem reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference : snapshots of Wikipedia from wikipedia.org) Example of a four colour map 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 four-color 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
Thoughts Arguments And Rants: Four Colours? from what i understand, four color theorem is a theorem in graph theory, ratherthan concerning any real graphs. the graphs that hudson uses in his http://tar.weatherson.net/archives/004335.html
Extractions: Donate to Oxfam: Australian link New Zealand link UK link US link ... Main Richard Zach reports that Georges Gonthier has a paper verifying the four colour map theorem. I found this odd, since I thought that Hud Hudson had shown that the theorem is not actually true, at least not as typically stated. Hudson's proof is here though that link may not be accessible to everyone. TrackBack Comments Might be worth mentioning that Melvin Fittingpretty well-known logicianalso claims that the four color theorem has been proven. Anyway here is the link/address: http://comet.lehman.cuny.edu/fitting/ Posted by: Mike at April 21, 2005 02:52 PM Well, what they prove is the theorem that every planar graph can be 4-colored. Hudson shows that it depends on how you define "simple planar map", "adjacent" etc. whether this is equivalent to "every simple planar map can be colored by 4 colors so that no two adjacent regions are colored by the same color." Hudson's example, for instance, corresponds not to a planar graph but to the complete graph on 6 vertices. The paper by Gonthier gives some explanation of how you have to understand "simple" and "adjacent" so that the correspondence holds (and these definitions exclude, I assume, the example Hudson constructs). Posted by: Richard Zach at April 21, 2005 05:16 PM
Citebase - Spiral Chains: A New Proof Of The Four Color Theorem Henceforth this paper offers another proof to the four color theorem which is not In this paper we have shown without assuming the four color theorem of http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:math/0408247
Citebase - A Simple Proof Of The Four-Color Theorem A simpler proof of the four color theorem is presented. The proof was reachedusing a series of equivalent theorems. First the maximum number of edges of a http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:math/0207061
Math G Mission College Santa Clara In 1890 The four Color theorem, once again, became the four Color He was amajor contributor to the four Color theorem and itís eventual proof. http://www.missioncollege.org/depts/math/beard2.htm
Extractions: Math Department, Mission College, Santa Clara, California Go to Math Dept Main Page Mission College Main Page This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material from this paper, please acknowledge it. To explore other such papers go to the Math G Projects Page. How many colors are required to color any map so that no countries with common borders are the same color? It is generally held that four colors, for any flat map, will suffice. But a belief that is commonly held and easily observed, is not a mathematical certainty. Nor does the simplicity of a question reflect the ease with which the answer can be proven. The mathematical evidence to create a valid proof that four colors are all that is required had evaded mathematicians for nearly 140 years. What became known as the Four Color Conjecture has been the cause of great fascination and frustration. It has also been the stimulus for new ideas in topology, knot theory, and the concept of mathematical proof. The question was originally posed by Francis Guthrie, a former student of the famous mathematician Augustus De Morgan, in 1852. Although Francis moved on to study law, his brother Frederick Guthrie had become a student of De Morgan. Francis Guthrie presented his work on the idea to his brother asking that he pass it along to De Morgan.
Absolute Certainty? proofs might become a useful method for verifying mathematical propositionsand large computationssuch as those leading to the four-color theorem. http://www.fortunecity.com/emachines/e11/86/certain.html
Extractions: "VIDEO PROOF" [See Video N25] dramatizes a theorem, proved by William P. Thurston of the Mathematical Sciences Research Institute (left), that establishes a profound connection between topology and geometry. The theorem shows how the space surrounding a complex knot (represented by the lattice in this scene) yields a "hyperbolic" geometry, in which parallel lines diverge and the sides of pentagons form right angles.The computer- generated video, called Not Knot, was produced at the Geometry Center in Minnesota. Computers are transforming the way mathematicians discover,prove and communicate ideas,but is there a place for absolute certainty in this brave new world? Legend has it that when Pythagoras and his followers discovered the theorem that bears his name in the sixth century B.C., they slaughtered an ox and feasted in celebration. And well they might. The relation they found between the sides of a right triangle held true not sometimes or most of the time but always-regardless of whether the triangle was a piece of silk or a plot of land or marks on papyrus. It seemed like magic , a gift from the gods. No wonder so many thinkers
Encyclopedia: Four-color-theorem The four color theorem states that any plane separated into regions, The fourcolor theorem was the first major theorem to be proven using a computer, http://www.nationmaster.com/encyclopedia/Four_color_theorem
Extractions: What's new? Our next offering Latest newsletter Student area Lesson plans Recent Updates List of religions List of largest wikis List of internet slang List of hooded figures ... More Recent Articles Top Graphs Richest Most Murderous Most Taxed Most Populous ... More Stats Updated 114 days 21 hours 4 minutes ago. Other descriptions of Four-color-theorem Example of a four color map The four color theorem states that any plane separated into regions, such as a political map of the counties of a state, can be colored using no more than 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. Each region must be contiguous - that is it may not be partitioned as are Michigan and Azerbaijan graphic for 4 color map theorem; by me for wiki File links The following pages link to this file: Four color theorem Categories: GFDL images ... graphic for 4 color map theorem; by me for wiki File links The following pages link to this file: Four color theorem Categories: GFDL images ...
Untitled Shortly after proofs of the fourcolours theorem were published by Arthur Bray We are now safe to say so, since the four-colours theorem was finally http://www.mimuw.edu.pl/delta/delta7/mapy/mapy.htm
Extractions: In 1852 a student of the University College in London, Francis Guthrie, informed his brother Frederick that, to his knowledge, any map can be painted with only four colours in such a way that no two neighbouring countries (i.e. which have a common frontier segment) share the same colour. Indeed, as Francis had noticed, four colours are sufficient to paint the counties on the map of England, so why wouldn't they be sufficient in all other cases, for arbitrary maps drawn on paper (i.e. on the plane) or on a globe (i.e. on a sphere)? To make things precise, let's assume the frontier of each country consists of one closed curve (which is not always the case with modern states). Frederick Guthrie passed the problem on to one of his teachers, Augustus de Morgan. De Morgan proved that there is no (plane) map with five pairwise neighbouring countries. Unfortunately, this is not enough to prove the non-existence of a map requiring at least five colours (see Fig. 1) and it is not clear whether de Morgan was conscious of this gap, when he mentioned the question to Hamilton in a letter of October 23 rd
Joho The Blog: Four Color Maps In 3D The air borders (in 3D) every country on the four color map, so you need at leastfive colors for the map and the air. You could then take a piece of silly http://www.hyperorg.com/blogger/mtarchive/001505.html
Extractions: An Entry from the Archives The W Resume Back to Blog ... A Note to Spammers May 15, 2003 Four Color Maps in 3D There was an interesting the Boston Globe recently about Four Colors Suffice , a book by Robin Wilson on the history of the famous 4-color problem: How do you prove that you only need four colors to ensure that neighboring countries are colored differently. (More important: Why is Greenland pink?) The proof (according to the article about the book that I didn't read) was the first generated by a computer that couldn't be checked by humans: in 1976, a Cray ground through every conceivable variation and found none that required more than four colors. I have a question for the mathematically inclined (i.e., people unlike me): How many colors would you need for a 3-D map? Or, if you prefer, how many colors would you need to ensure that blocks (of any shape) stacked in any arbitrary way have differently colored neighbors? I am so bad at 3D stuff that you could tell me the answer is 2 and I would believe you, just so long as you looked at me with those doe-eyes of yours. Posted by D. Weinberger at May 15, 2003 09:35 AM
