Advances in the use of Systems of Equalities and Inequalities to an analytical demonstration of the Four-Color Theorem.

Abstract

The Four-Color Theorem was first solved using computational methods in 1977, after 125 years of attempts. The only currently accepted approaches are those derived from the original paper by Appel and Haken. Although efforts to solve this problem have led to the development of new branches of mathematics, an analytical proof remains elusive. A simple internet search reveals hundreds of unrecognized attempts at analytical solutions proposed by numerous researchers. This article presents a self-contained revision of the most recent publication by the author on this topic. The author maintains that it is possible to translate the Four-Color Theorem into a system of equalities and inequalities. In the previous article, the starting point was to reinterpret the problem as one of coloring “countries” distributed over a spherical surface. Spherical coordinates were employed to map the surface onto a plane. The concept of a hyperline was introduced and added to this plane, and a corresponding system of algebraic equalities and inequalities was established. In the present article, these concepts are revisited. Additional considerations regarding the algebraic system is developed, and the map-coloring problem is reformulated as a problem of coloring segments of the corresponding hyperline of a map. The resulting system is solved, offering a simple, clear, and elegant demonstration of the Four-Color Theorem. As this work is a revision of a single previous article, all other references have been intentionally omitted.