A Search of The Four-Color Theorem and its Higher Dimensional Generalization

TL;DR

This paper presents a proof of the Four-Color Theorem using Kuratowski's Theorem and induction, describes the most complex coloring map, offers a simple proof of Kuratowski's Theorem via Euler characteristic, and conjectures its higher-dimensional generalization.

Contribution

It provides a new proof of the Four-Color Theorem based on Kuratowski's Theorem and introduces a conjecture for its higher-dimensional extension.

Findings

Proof of Four-Color Theorem using Kuratowski's Theorem and induction

Description of the most complicated coloring map

Simple proof of Kuratowski's Theorem via Euler characteristic

Abstract

Four-Color Theorem has secret in its logical proof and actual operating. In this paper we will give a proof of Four-Color Theorem based on Kuratowski's Theorem using some induction argument and give a description of the most complicated coloring map, a simple proof of Kuratowski's Theorem using Euler charateristic is also presented. We also conjecture the higher dimensional generalization of Four-Color Theorem.