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The four color theorem
The Four Color Theorem states that for any subdivision of a plane into adjacent regions, the regions can be colored with no more than four colors such that no two adjacent regions share the same color. The theorem was first proven in 1976 by Kenneth Appel and Wolfgang Haken using a computer, which was the first major theorem to be proven this way. It took a long time to prove because finding counterexamples that disproved previous attempts was difficult, as it required checking all possible colorings. The proof works by reducing configurations to a small set that can be colored with four colors, reaching a contradiction if a map requires more colors.
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The four color theorem
- 1. The Four Color Theorem Forany subdivision of the plane into nonoverlapping regions, it is always possible to mark each of the regions with four different colors in such a way that no two adjacent regions receive the same color .
- 2. What is theFour Color Theorem? Given any separation of a plane into adjacent regions , the regions can be colored using at most four colors so that no two adjoining regions have the same colour A region is adjacent to the other region if they share a boundary and not just a point.
- 3. HISTORY The four colortheorem was proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallestsized counterexample to the four color theorem. Appel and Haken used a special-purpose computer program to confirm that 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. Showing 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 that the theorem is therefore 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 Appel–Haken 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.
- 4. Why it tookso long to prove the Theorem? The Four Color Problem It was easy to prove that 6 colors would work and reduce that numbers to 5 but proving that 4 colors would work took longer and many counterexamples exist. Most counter examples try to have one region that is adjacent to all other regions. No way to check with actually coloring the maps.
- 5. Proving the FourColor Theorem The proof relies on being able to reduce a set of configurations on a graph. This small configuration of the original can be easily colored with four colors. Many times the figures can be colored by focusing on one region and trying to color the figures around it with the other colors. To prove the (network version of the) Four Color Theorem, you start out by assuming that there is a network that cannot be colored with four colors, and work to deduce a contradiction. If there is such a network, there will be (at least) one that has the fewest number of nodes. That's the one to look at. The idea then is to show that you can find a particular node that can be removed without altering the number of colors needed to color the network. Since that new network has one fewer nodes than the one you started with, and that initial network was chosen to be the smallest that could not be colored with four colors, the new network can be colored with four colors. But then, because of the way you chose the node to remove, that means the original map can be colored with four colors. And there's the contradiction.
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