A DISCRETE METHOD TO SOLVE THE FOUR COLOR THEOREM

Abstract

The Four-Color Theorem originated from the attempt to solve the problem of painting MAPS over a plane or spherical surface. Over a century and a half, this problem underwent various abstractions until it was resolved in 1976. The proposed solution, which is disruptive, computationally calculates the number of possible states for a representation of a flat map. Although it is resolved, the lack of a formal proof for this problem causes some discomfort. Therefore, a solution that uses more traditional techniques and is easily understandable is needed. In a previous article, a solution based on equalities and inequalities between boundaries was presented. Now in this article, a generic spheroidal MAP is subjected to various one-to-one relationships until a generator of all possible MAPS on a two-dimensional surface partitioned into n² cells are found. Four-Colors are proved to be necessary and sufficient to paint a two dimensional MAP. It is explained at the end of the article that the imposition of a fifth color as a necessary condition implies a contradiction.