A Victorian Age Proof of the Four Color Theorem

TL;DR

This paper revisits the four color theorem, providing a new method to rule out counterexamples and presenting an algorithmic proof based on face coloring of cubic planar maps.

Contribution

It introduces a novel approach to eliminate counterexamples to Kempe's proof and offers an algorithmic proof using face coloring techniques.

Findings

Counterexamples to Kempe's proof can be constructed and ruled out.

A three-step algorithmic proof achieves four-coloring of cubic planar maps.

The method simplifies four-coloring by avoiding odd cycles in uncolored regions.

Abstract

In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be rule out by re-constructing special 2-colored two paths decomposition in the form of a double-spiral chain of the maximal planar graph. In the second part of the paper we have given an algorithmic proof of the four color theorem which is based only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has been obtained almost trivially.