A Study on Hand Proof for The Four-Color Theorem

TL;DR

This paper presents a novel hand proof for the four-color theorem, introducing creation and annihilation operations to demonstrate how four colors suffice for map coloring and explaining the roles of trapped and boundary vertices.

Contribution

It offers a new hand proof approach using creation and annihilation operations, providing insights into the mechanisms behind the four-color theorem.

Findings

Four colors are sufficient to color any map.

Identification of trapped and boundary vertices influences coloring.

A method to iteratively color maps using creation and annihilation operations.

Abstract

For the four-color theorem that has been developed over one and half centuries, all people believe it right but without complete proof convincing all1-3. Former proofs are to find the basic four-colorable patterns on a planar graph to reduce a map coloring4-6, but the unavoidable set is almost limitless and required recoloring hardly implements by hand7-14. Another idea belongs to formal proof limited to logical operation15. However, recoloring or formal proof way may block people from discovering the inherent essence of a coloring graph. Defining creation and annihilation operations, we show that four colors are sufficient to color a map and how to color it. We find what trapped vertices and boundary-vertices are, and how they decide how many colors to be required in coloring arbitrary maps. We reveal that there is the fourth color for new adding vertex differing from any three coloring vertices in creation operation. To implement a coloring map, we also demonstrate how to color an arbitrary map by iteratively using creation and annihilation operations. We hope our hand proof is beneficial to understand the mechanisms of the four-color theorem.