What is four colour theorem .
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Answer:
In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color.
Step-by-step explanation:
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Step-by-step explanation:
In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet.[1] It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand.[2] Since then the proof has gained wide acceptance, although some doubters remain.[3]
The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map). To dispel any remaining doubts 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 proved 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 Three-coloring
6 Generalizations
7 Relation to other areas of mathematics
8 Use outside of mathematics
9 See also
10 Notes
11 References
12 External links
Precise formulation of the theorem
In graph-theoretic terms, the theorem states that for loopless planar graph {\displaystyle G}G, the chromatic number of its dual graph is {\displaystyle \chi (G^{*})\leq 4}{\displaystyle \chi (G^{*})\leq 4}.
The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.
First, regions are adjacent if they share a boundary segment; two regions that share only isolated boundary points are not considered adjacent. Second, bizarre regions, such as those with finite area but infinitely long perimeter, are not allowed; maps with such regions can require more than four colors.[4] (To be safe, we can restrict to regions whose boundaries consist of finitely many straight line segments. It is allowed that a region entirely surround one or more other regions.) Note that the notion of "contiguous region" (technically: connected open subset of the plane) is not the same as that of a "country" on regular maps, since countries need not be contiguous (e.g., the Cabinda Province as part of Angola, Nakhchivan as part of Azerbaijan, Kaliningrad as part of Russia, and Alaska as part of the United States are not contiguous). If we required the entire territory of a country to receive the same color, then four colors are not always sufficient. For instance, consider a simplified map: