A Pure Mathematical Proof of 4-Colour Theorem

Abstract

This work "A Pure Mathematical Proof of the 4-Colour Theorem" is related to the previous proof assisted by computer. "Triangulations of Euler Convex Polygon" provides a fresh beginning point for the proof. The central concept is to discover an extended invariance property of Standard Graph’s boundary, which is described as "3-Colour All Phase States (3CP)" in this work and it is demonstrated that the Standard Graph’s boundary and sub-bound are 3CP and 4-colorable (4-3CP) via the Expanded Operation e(+, pi) and e(-, pi). It's exciting that this regularity was discovered for the first time and the 4-3CP invariance can naturally derive the 4-Colour Theorem. The majority of the definitions, theorems and proof strategies are shown in this work.