A Constructive Proof of the Four-Color Theorem

TL;DR

This paper offers a new constructive proof of the Four-Color Theorem by developing a framework that incrementally colors planar graphs without relying on computer enumeration.

Contribution

It introduces novel concepts and a framework that systematically proves the theorem through constructive, step-by-step boundary modifications.

Findings

Framework guarantees four-colorability after boundary modifications

Avoids computer enumeration in the proof process

Provides a constructive perspective on the Four-Color Theorem

Abstract

This paper presents a path to proving the Four-Color Theorem that differs from the traditional "reducible configuration" method. By introducing concepts such as "outer boundary," "primitive set," "Property A," "knot," "valid pair group," and the operation of "adding an n-point region on an interval," we construct a framework for gradually coloring any given planar graph. The core of this framework consists of three theorems, which ensure that after sequentially adding specific regions on an outer boundary satisfying Property A, the new outer boundary still satisfies Property A, ultimately allowing the entire given graph to be colored with four colors. This method avoids computer enumeration and provides a more constructive proof perspective.