Use of Equalities and Inequalities System to demonstrate the Four Color Theorem

Resumo

The Four-Color Theorem (T4C) originated from the attempt to resolve the challenge of coloring maps over a plane or spherical surface. Over a century and a half, this problem underwent various abstractions until finding resolution in 1976. The proposed solution, which is groundbreaking, computationally calculates the multitude of representations of flat maps. Despite its resolution, the absence of a formal proof for this problem using traditional mathematical tools induces some unease, and a readily comprehensible solution is desirable. Nowadays, numerous articles propose diverse solutions to this problem but none of which have garnered official recognition. Taking a distinctive approach, solutions grounded in function properties and infinitesimal discretization were presented by this author in 2021 and 2023.

This article introduces an infinite covering of a map by a set of parallel lines, effectively capturing the structure of the map. A series of segmented straight lines are generated, and it is demonstrated that they are paintable with Four Colors, implying that the corresponding maps are colorable by Four Colors as well.