![]() Forcing two separate regions to have the same color can be modelled by adding a 'handle' joining them outside the plane. If we wanted those regions to receive the same color, then five colors would be required, since the two A regions together are adjacent to four other regions, each of which is adjacent to all the others. In this map, the two regions labeled A belong to the same country. If we required the entire territory of a country to receive the same color, then four colors are not always sufficient. It is allowed that a region has enclaves, that is it entirely surrounds 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 (they may have exclaves, e.g., the Cabinda Province as part of Angola, Nakhchivan as part of Azerbaijan, Kaliningrad as part of Russia, France with its overseas territories, and Alaska as part of the United States are not contiguous). ![]() (To be safe, we can restrict to regions whose boundaries consist of finitely many straight line segments. (Otherwise, a map in a shape of a pie chart would make an arbitrarily large number of regions 'adjacent' to each other at a common corner, and require arbitrarily large number of colors as a result.) 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. 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.įirst, regions are adjacent if they share a boundary segment two regions that share only isolated boundary points are not considered adjacent. ![]() In graph-theoretic terms, the theorem states that for loopless planar graph G. In 2005, the theorem was also proved by Georges Gonthier with general-purpose theorem-proving software. 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. 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). The proof has gained wide acceptance since then, although some doubters remain. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. It was the first major theorem to be proved using a computer. Adjacent means that two regions share a common boundary curve segment, not merely a corner where three or more regions meet. In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. Statement in mathematics Example of a four-colored map A four-colored map of the states of the United States (ignoring lakes and oceans) ![]()
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