Is the Four-Color Theorem a Real Analysis problem?

The Four-Color Theorem (T4C) was conjectured in 1852 and states that any planar map can be colored with four colors such that no two adjacent regions have the same color. After 125 years of attempts, the theorem was finally proven in 1977 using computational methods. All currently accepted proofs are derivatives of the original work by Appel and Haken. Despite the fact that efforts to solve this problem led to the development of new branches of mathematics, an analytical proof remains undefined. Searching for an analytical demonstration this paper re-frames the four-color problem as a Real Analysis problem in the R2 plane. The information contained in MAPs are expressed as a System of Equalities and Inequalities as previously did by Jansen J. U. The use of rudimentary concepts of Real Analysis and Combinatorial Analysis reduces the complexity of the analysis and conducts to a general solution of a MAP with any number of colors. This article is an improvement of a previous one written by this author with only this one being intentionally cited.

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Published
2025-09-01
DOI
https://doi.org/10.1590/scielopreprints.13019
Primary Topic
Color Science and Applications
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article
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article

Is the Four-Color Theorem a Real Analysis problem?

José Ulisses Jansen
Color Science and Applications
article

Is the Four-Color Theorem a Real Analysis problem?

José Ulisses Jansen
article en

Abstract

The Four-Color Theorem (T4C) was conjectured in 1852 and states that any planar map can be colored with four colors such that no two adjacent regions have the same color. After 125 years of attempts, the theorem was finally proven in 1977 using computational methods. All currently accepted proofs are derivatives of the original work by Appel and Haken. Despite the fact that efforts to solve this problem led to the development of new branches of mathematics, an analytical proof remains undefined. Searching for an analytical demonstration this paper re-frames the four-color problem as a Real Analysis problem in the R2 plane. The information contained in MAPs are expressed as a System of Equalities and Inequalities as previously did by Jansen J. U. The use of rudimentary concepts of Real Analysis and Combinatorial Analysis reduces the complexity of the analysis and conducts to a general solution of a MAP with any number of colors. This article is an improvement of a previous one written by this author with only this one being intentionally cited.

Universidade Estadual de Campinas (UNICAMP) (BR)
Openalex Percentile: Top 82%
Color Science and Applications
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