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Product Features and Technical DetailsProduct Features2.0 GHz Intel Core Duo processor with 2 MB shared L2 Cache 1 GB (single SODIMM) 667 MHz DDR2 SDRAM (PC2-5300); 100 GB 5400 rpm Serial ATA hard drive; slot-load SuperDrive (DVD隆脌RW/CD-RW) One FireWire 400, two USB 2.0 ports, and ExpressCard/34 slot; no FireWire 800 slots Built-in 10/100/1000BASE-T (Gigabit); built-in 54 Mbps AirPort Extreme (802.11g); built-in Bluetooth 2.0+EDR 15.4-inch TFT widescreen display with 1440 x 900 resolution Processor, Memory, and Motherboard Hardware Platform: Mac Processor: 2 GHz Intel Core Duo System Bus Speed: 667 Number of Processors: 1 RAM: 1 GB RAM Type: DDR2 SDRAM Hard Drive Size: 100 GB Manufacturer: Portable Type: Serial ATA Graphics and Display Graphics RAM: 256 MB Ports and Connectivity Modem: None Cases and Expandability Size (LWH): 19.5 inches, 7.6 inches, 19.5 inches Weight: 5.6 pounds Product Description
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Example of a four-colored map
A four-coloring of an actual map of the states of the United States (ignoring water and other countries).
In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color. Two regions are called adjacent only if they share a border segment, not just a point.
Three colors are adequate for simpler maps, but an additional fourth color is required if one region is surrounded by three regions that touch each other. The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century; however, proving four colors suffice turned out to be significantly harder. A number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852.
Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers. According to an article by the math historian Kenneth May (Wilson 2002, 2), aps utilizing only four colours are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property.
The four color theorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proven using a computer. Appel and Haken's approach started by showing there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to check each of these maps had this property. Additionally, any map (regardless of whether it is a counterexample or not) must have a portion that looks like one of these 1,936 maps. To show this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples existed because any must contain, yet not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and the theorem is true. Initially, their proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand (Swart 1980). Since then the proof has gained wider acceptance, although doubts remain (Wilson 2002, 216-222).
To dispel remaining doubt about the Appelaken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour and Thomas. Additionally in 2005, the theorem was proven by Georges Gonthier with general purpose theorem proving software.
Contents
1 Precise formulation of the theorem
2 History
2.1 Early proof attempts
2.2 Proof by computer
2.3 Simplification and verification
3 Summary of proof ideas
4 False disproofs
5 Generalizations
6 See also
7 References
8 External links
//
Precise formulation of the theorem
The intuitive statement of the four color theorem needs to be interpreted appropriately to be correct. For example, each region of the map should be contiguous.
Example of a map with non-contiguous regions
In the real world, not all countries are contiguous (e.g., Brunei, Alaska as part of the United States, Nakhchivan as part of Azerbaijan, and Kaliningrad as part of Russia). Because the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map:
In this map, the two regions labeled A belong to the same country, and must be the same color. This map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. If A consisted of three regions, six or more colors might be required; one can construct maps that require an arbitrarily high number of colors.
Some clarification is also needed on when two regions are said to be adjacent. For the theorem to be correct, two regions should only be considered adjacent if they share a nonzero length of boundary; touching at a single boundary point (such as the point at Four Corners where Arizona, Colorado, New Mexico, and Utah meet) does not count as an adjacency for the purposes of the theorem.
To precisely state the theorem, it is easiest to rephrase it in graph theory. It then states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or "every planar graph is four-colorable" for short (Thomas 1998; Wilson 2002). Such a graph can be obtained from a map by replacing every region by a vertex, and connecting two vertices by an edge exactly when the two...(and so on)
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