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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.12 The Four-Color Theorem

Bojan Mohar

The four-color theorem asserts that the regions of any map drawn in the plane (or, equivalently, on the two-dimensional sphere) can be colored with no more than four colors in such a way that any two regions with a common boundary are given different colors. The example in figure 1 shows that four distinct colors are necessary since the regions A, B, C, and D are all adjacent to each other. This result was conjectured by Francis Guthrie in 1852. An incorrect proof was given by Kempe in 1879, and for eleven years the problem was believed to have been solved, until Heawood pointed out the error in 1890. However, Heawood showed that Kempe’s basic idea, which we shall outline below, could at least be used ...

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