Mathematical Proofs of Two Conjectures: The Four Color Problem and The Uniquely 4-colorable Planar Graph

TL;DR

This paper presents a purely mathematical proof of the four color theorem and the uniquely 4-colorable planar graph conjecture, offering an alternative to computer-assisted proofs and simplifying the understanding of these longstanding problems.

Contribution

The paper introduces a new mathematical proof method for the four color theorem and the uniquely 4-colorable planar graph conjecture, reducing reliance on computational verification.

Findings

Proves the four color theorem using only 4 reducible unavoidable sets

Provides a mathematical proof for the uniquely 4-colorable planar graph conjecture

Simplifies the proof process compared to previous computer-assisted methods

Abstract

The famous four color theorem states that for all planar graphs, every vertex can be assigned one of 4 colors such that no two adjacent vertices receive the same color. Since Francis Guthrie first conjectured it in 1852, it is until 1976 with electronic computer that Appel and Haken first gave a proof by finding and verifying 1936 reducible unavoidable sets, and a simplified proof of Robertson, Sanders, Seymour and Thomas in 1997 only involved 633 reducible unavoidable sets, both proofs could not be realized effectively by hand. Until now, finding the reducible unavoidable sets remains the only successful method to use, which came from Kempe's first "proof" of the four color problem in 1879. An alternative method only involving 4 reducible unavoidable sets for proving the four color theorem is used in this paper, which takes form of mathematical proof rather than a computer-assisted proof and proves both the four color conjecture and the uniquely 4-colorable planar graph conjecture by mathematical method.