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Purposing an Algebraic Solution to the Four-Color Problem




Four-Color Theorem, Complex deployment


The Four-Color Theorem was originated with the coloring of Countries in a MAP and it was a challenging problem that remained open since 1853 for more than 170 years. By the end of Sec XX, this problem was solved using computational tools but until today there is no algebraic proof of it. In this article, the original problem of coloring MAPS over a Spherical Surface is briefly reviewed. A Spherical MAP is converted into a Planar MAP using polar coordinates and the frontiers of the Countries are described as real implicit equations and then deployed from the real space into the complex space. In the complex space the rules to color MAPs are described as system of algebraic equations and inequations. One example of MAP is solved (colored) and the explanation about why these systems are solvable is done. Beginning from the example, a general theory to coloring MAPs is derived. As all the transformations used admits inverse, the obtained planar MAP solution can be reversed as a solution to the Spherical MAP. All operations involve simple algebraic transformations and some Calculus concepts


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Exact and Earth Sciences