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Topic: Four Color

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	Encyclopedia: Four color theorem
		The four color theorem was the first major theorem to be proved using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof).
		The four color theorem states that given any finite plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color.
		The four color theorem was the first major theorem to be proved using a computer, and the proof is not accepted by all mathematicians because it would be infeasible for a human to verify by hand.
	www.nationmaster.com /encyclopedia/Four_color_theorem (753 words)

	Four color theorem - Wikipedia, the free encyclopedia
		Textbooks on cartography and the history of cartography don't mention the four color theorem, even though map coloring is a subject of discussion.
		Perhaps one effect underlying this common misconception is the fact that the color restriction is not transitive: a region only has to be colored differently from regions it touches directly, not regions touching regions that it touches.
		This was initially known as the Heawood conjecture and proved as The Map Color Theorem by Gerhard Ringel and J.
	en.wikipedia.org /wiki/Four_color_theorem (2096 words)

	Four-color Printing Encyclopedia Article, Information, History and Biography @ ColorFool.com (Site not responding. Last check: 2007-10-14)
		Color printing is the reproduction of an image or text in color (as opposed to simpler fl and white or monochrome printing).
		Another emerging method of color printing is six-color process printing (for example, Pantone's Hexachrome system) which adds orange and green to the traditional CMYK for a larger and more vibrant gamut, or color range.
		Due to limitations in the ink pigments, the darker colors are dirty and muddied.
	www.colorfool.com /encyclopedia/Four-color_printing (1179 words)

	Ivars Peterson's MathLand
		The question is whether four colors are always enough to fill in every conceivable map that can be drawn on a flat piece of paper so that no countries sharing a common boundary are the same color.
		Because the conjecture that four colors suffice hadn't yet been proved, Carroll didn't know with certainty whether the answer to his question was four or five.
		Four colors turn out to be necessary in any situation in which a region has common borders with an odd number of neighboring regions.
	www.maa.org /mathland/mathland_1_6.html (1007 words)

	Read about Four color theorem at WorldVillage Encyclopedia. Research Four color theorem and learn about Four color ... (Site not responding. Last check: 2007-10-14)
		colors in such a way that no two adjacent regions receive the same color.
		Maps utilizing only four colors are rare, and those that do usually require only three.
		Textbooks on cartography and the history of cartography don't mention the four colour theorem, even though map colouring is a subject of discussion.
	encyclopedia.worldvillage.com /s/b/Four_color_theorem (1217 words)

	How Four Color Printing Works
		Color reproduction is based on the theory of three color vision.
		These three colors are called additive primaries because three lights of these three colors, when added together, produce white light.
		A fourth color, fl, is added to the mix to overcome this, improving the shadow and contrast of the image.
	www.intaglio.com /4color.html (276 words)

	Mappa.Mundi Magazine - Locus - The Four-Color Map Problem
		It is possible to go to Four Corners and put one hand or foot in each of the states, but on a map two diagonally opposed states could be the same color.
		Bridgman's, 1896 Rail Road and Township map of New York illustrates the four color mapping problem - as a practical matter, green, yellow, pink and tan are sufficient to map the townships.
		Some say the four color theorum was finally proved by Appel and Haken in 1976, but others claim that the question is yet to be resolved satisfactorily.
	mappa.mundi.net /locus/locus_014 (873 words)

	Learn more about Four color theorem in the online encyclopedia. (Site not responding. Last check: 2007-10-14)
		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.
		This theorem was conjectured in 1853 by Francis Guthrie.
		Five colors are required if those two regions are to receive the same color.
	www.onlineencyclopedia.org /f/fo/four_color_theorem.html (682 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-14)
		The Four Color Problem is one of my favorites because it is the first unsolved (it was at the time) problem that I was ever introduced to.
		To color a map, you must assign each region a color and no two regions may have the same color if they share a side (one point doesn't count).
		Date: Thu, 8 Dec 1994 22:03:40 -0500 From: Stephen Weimar Subject: Re: Four color Map Problem The four-color map theorem was proved by Appel and Haken at the University of Illinois at Urbana-Champaign in 1976.
	www.forum.swarthmore.edu /dr.math/problems/4_color_map.html (392 words)

	Colorful Mathematics/Teacher
		However, using only four colors is not as straightforward as it may appear so six colors have been provided which should make it relatively easy for everyone to color their maps correctly.
		After having completely colored a map, a message will appear on the screen asking the student to try to improve on the number of colors required, or to be able to show that this number cannot be reduced.
		The use of one color to select some vertices in a graph so that every vertex is either colored or connected by an edge to a colored one creates a dominating set of a graph.
	www.math.ucalgary.ca /~laf/colorful/teacher.html (2018 words)

	The Four Color Theorem (Site not responding. Last check: 2007-10-14)
		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 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.
		This was confirmed by Appel and Haken in 1976, when they published their proof of the Four Color Theorem [1,2].
	www.math.gatech.edu /~thomas/FC/fourcolor.html (1878 words)

	Graphics Color Models
		A color model is an orderly system for creating a whole range of colors from a small set of primary colors.
		Since a color model has a particular range of available color or gamut, it is referred to as using a "color space".
		In the RGB color model, colors are represented by varying intensities of red, green and blue light.
	www.sketchpad.net /basics4.htm (1228 words)

	CJPW Beyond Four-Color Process
		Your four color process is normally printed on a five or six-color press, which leaves one or two "units" open for additional colors or finishes.
		Use of the fifth unit in combination with the other four "frames" is economical, since you are essentially "piggy-backing" this fifth color on the same run as the first four colors.
		With the process color dry from the first run, the "off-line" varnish is laid down on a sealed surface, so less ink is absorbed by the paper stock.
	www.cjpw.com /beyond.htm (983 words)

	The Four Color Theorem
		The coloring of geographical maps is essentially a topological problem, in the sense that it depends only on the connectivities between the countries, not on their specific shapes, sizes, or positions.
		Likewise the four vectors assigned to the vertices of a tetrahedral graph would be a complete set of four mutually orthogonal vectors (a tetrad), but still with arbitrary orientation.
		In this context the four color theorem tells us that a space of four dimensions is sufficient to enable us to assign one of the four basis vectors to each vertex of a planar graph in such a way that the vectors of every pair of adjacent vertices are orthogonal.
	www.mathpages.com /home/kmath266/kmath266.htm (4081 words)

	Four Color Theorem Encyclopedia Article, Information, History and Biography @ SomeLocalColor.com (Site not responding. Last check: 2007-10-14)
		Looking For four color theorem - Find four color theorem and more at Lycos Search.
		Find four color theorem - Your relevant result is a click away!
		Look for four color theorem - Find four color theorem at one of the best sites the Internet has to offer!
	www.somelocalcolor.com /encyclopedia/Four_color_theorem (2248 words)

	Custom Four-Color Process
		White text should be created with a fifth color of white (Additional charges will apply) or reversed from a single solid color.
		Remember: White is a color on transparent decals.
		Color Match: Add $30.00 per color to be matched other than our standard colors.
	www.needdecals.com /fourcolorprocess.html (229 words)

	U Printing: A Full Color Printing Company
		Color is known increases customer response rates by a whopping 80%.
		A variety of great printing solutions are catered to suit the specifications of clients regarding the fulfillment of significant projects, whether it is for digital printing, graphic printing, 4 color process printing, press printing, custom printing, large format poster printing or even mailing services and greeting cards printing.
		With regard to superior printing capabilities in the area of 4 color process printing, large format poster printing, and digital printing, we make use of up to date software systems along with variable data printing, discount printing and custom printing facilities for better-quality output.
	www.uprinting.com (876 words)

	CJPW Four-Color Printing
		The four basic process colors -- cyan, magenta, yellow and fl, as almost everyone in the printing or graphic business knows only too well, when combined and printed with great care and artistry, can yield very gratifying collateral materials.
		There's a message there, in the words "great care and artistry", and that about sums up what we at CJPW feel for every job that passes our desks - whether it be four color process or two color or three color or whatever.
		To assure our customers of the highest possible quality, especially with four color process, our considerations extend to many areas, including how the equipment is maintained, the condition of water and ink rollers, humidity controls in the plant, use of the highest quality inks and so on.
	www.cjpw.com /4color.htm (447 words)

	What is color separation? - A Word Definition From the Webopedia Computer Dictionary
		This type of color separation, mixing three or four colors to produce an infinite variety of colors, is called process color separation.
		Another type of color separation, called spot color separation, is used to separate colors that are not to be mixed.
		Spot colors are effective for highlighting text but they cannot be used to reproduce full-color images.
	www.webopedia.com /TERM/C/color_separation.html (270 words)

	The Four Color Problem and its connection to South African Flora
		He asked whether it is possible to color any map with four or fewer colors so that adjacent regions (those that share a common boundary) are colored differently.
		Awareness of the Four Color Problem increased substantially when, on June 13, 1878, the renowned mathematician Arthur Cayley asked if the problem had been solved.
		Shortly afterwards, Cayley [2] published a paper on the Four Color Problem, in which he postulated why this problem appears to be so difficult.
	www.uwinnipeg.ca /~ooellerm/guthrie/FourColor.html (703 words)

	Claim of Proof to Four-Color Theorem
		Our aim was to give the map in S1 a form that does not allow it more than four colours, and then to prove (if we could) that such a form will generate all maps.
		At this stage we had little doubt that the four colour conjecture was true, ard various mathematicians with whom we corresponded, including Bertrand Russell, were aware that we had a method that was almost certainly capable of proving it.
		Since the Americans have not published a proof, their announcement of course does not preempt the issue, and leaves the position in respect of the first published proof exactly as before.
	www.lawsofform.org /gsb/nature.html (688 words)

	Mudd Math Fun Facts: Four Color Theorem
		It is easy to show that you need at least four colors, because Figure 1 shows a map with four countries, each of which is touching the other.
		The four color theorem is true for maps on a plane or a sphere.
		The answer is different for geographic maps on a torus; it turns out that 7 colors is necessary and sufficient then...
	www.math.hmc.edu /funfacts/ffiles/10003.4.shtml (270 words)

	Colorful Mathematics/Games/4colors (Site not responding. Last check: 2007-10-14)
		This game is based on a simple idea: draw a map or any picture as complicated as you wish and color each region using the fewest possible number of colors, the only requirement being that regions sharing a common border must receive different colors.
		It took over a hundred years for mathematicians to prove that four colors were sufficient, no matter how complicated the map.
		When the map is completed, the computer will count the number of colors used and display one of a variety of messages evaluating the results.
	www.math.ucalgary.ca /~laf/colorful/4colors.html (417 words)

	Speedcolor - Commercial Printing Services
		Our quality is what you see in the very best magazines, Four Color catalogs and brochure printing promotions that arrive in your mail every day.
		Four Color (full color) promotional brochures, flyers, posters, mailers or cards that powerfully promote your organization's product or services.
		Four Color catalogs that make your product or services jump from the page with lasting appeal to your audience.
	www.speedcolor.com (432 words)

	The Most Colorful Math of All (Site not responding. Last check: 2007-10-14)
		Even at that time, it was common knowledge among map-makers that 4 colors seemed to be sufficient to color a map.
		Given any map, there is, however, a reasonably fast way to tell whether or not you can color it with two colors, and we do know you won't ever need more than four colors.
		Greedy Algorithms are one of the four or five most basic and general algorithm design strategies used in computer science.
	www.c3.lanl.gov /mega-math/workbk/map/mpbkgd.html (1177 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-10-14)
		The Four Color Problem was famous and unsolved for many years.
		The basic rule for coloring a map is that no two regions that share a boundary can be the same color.
		Mapmakers are not mathematicians, so the assertion that only four colors would be necessary for all maps gained acceptance in the map-making community over the years because no one ever stumbled upon a map that required the use of five colors.
	www.c3.lanl.gov /mega-math/gloss/math/4ct.html (541 words)

	Four Color Process Printing
		Four color process is a system where a color image is separated into 4 different color values by the use of filters and screens (usually done digitally).
		The result is a color separation of 4 images that when transferred to printing plates and printed on a printing press with the colored inks cyan (blue), magenta (Red), yellow and fl, reproduces the original color image.
		A university study showed a 65% increase in the retention of material when full color was used instead of fl and white.
	www.4bizcards.com /process.html (163 words)

	Bivariate Splines and the Four Color Map Problem
		This map of the United States was colored by Barrett Walls, in a project with Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas.
		The celebrated Four Color Map Theorem states that any map in the plane or on the sphere can be colored with only four colors such that no two neighboring countries are of the same color.
		The question of whether four colors always suffice was first stated in 1852 by Francis Guthrie and remained unanswered until Appel and Haken came up with a book length proof in 1976.
	www.math.utah.edu /%7Ealfeld/talks/S13/4CMP.html (429 words)

	Color business card design and business card printing - Andreoni.com
		Andreoni.com prints in four color process, on 13 or 15 pt card stock with professional printing presses, not digital printers.
		RGB files cannot be printed on professional four color presses.
		Not every RGB color has a CMYK equivalent, and the result could be more than just subtle color shifts.
	www.andreoni.com /services_businesscards.htm (905 words)

	Four Color Theorem (Site not responding. Last check: 2007-10-14)
		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.
		A summary of a new proof of the four color theorem is at Georgia Tech for which programs and data supplements are available by FTP.
		A recent reference to this is "Map Coloring and the Vector Cross Product" by Louis H. Kauffman in Journal of Combinatorial Theory B 48(1990)145-154.
	grail.cba.csuohio.edu /~somos/4ct.html (329 words)

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