Algorithmic methods of finite discrete structures. The Four Color Theorem. Theory, methods, algorithms

TL;DR

This paper explores a novel approach to the Four Color Theorem by reducing planar graphs to cubic graphs and applying algebraic transformations, aiming to both prove and provide algorithms for four-coloring planar graphs.

Contribution

It introduces a method based on reducing maximally flat graphs to cubic graphs and uses algebraic properties to develop coloring algorithms for planar graphs.

Findings

Reduction of maximally flat graphs to cubic graphs enables four-coloring.

Coloring cubic graphs can be achieved using Klein group transformations.

Algorithms for planar graph coloring are derived from algebraic properties.

Abstract

The Four color problem is closely related to other branches of mathematics and practical applications. More than 20 of its reformulations are known, which connect this problem with problems of algebra, statistical mechanics and planning. And this is also typical for mathematics: the solution to a problem studied out of pure curiosity turns out to be useful in representing real objects and processes that are completely different in nature. Despite the published machine methods for combinatorial proof of the Four color conjecture, there is still no clear description of the mechanism for coloring a planar graph with four colors, its natural essence and its connection with the phenomenon of graph planarity. It is necessary not only to prove (preferably by deductive methods) that any planar graph can be colored with four colors, but also to show how to color it. The paper considers an approach based on the possibility of reducing a maximally flat graph to a regular flat cubic graph with its further coloring. Based on the Tate-Volynsky theorem, the vertices of a maximally flat graph can be colored with four colors, if the edges of its dual cubic graph can be colored with three colors. Considering the properties of a colored cubic graph, it can be shown that the addition of colors obeys the transformation laws of the fourth order Klein group. Using this property, it is possible to create algorithms for coloring planar graphs.