TITLE:
4-Staining of “Staining Dilemma Configuration” in Four-Color Conjecture
AUTHORS:
Yudian Zhang, Lichong Zhang
KEYWORDS:
Four-Color Conjecture, Kempe Proof, Staining Dilemma Configuration, Four-Color Vertex Quadrilateral, Non-10 Fold Symmetric Transformation
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.10 No.3,
March
25,
2022
ABSTRACT: This article attempts to successfully fill Kempe proof loophole, namely 4-staining of “staining dilemma configuration”. Our method is as follows: 1) Discovered and proved the existence theorem of the quadrilateral with four-color vertices and its properties theorems, namely theorems 1 and 2. From this, the non-10-fold symmetry transformation rule of the geometric structure of Errera configuration is generated, and using this rule, according to whether the “staining dilemma configuration” is 10-fold symmetry, they are divided into two categories; 2) Using this rule, combining the different research results of several mathematicians on Errera graphs, and using four different classifications of propositional truth and falsehood, a new Theorem 3 is established; 3) Using Theorem 3, the theoretical proof that the non-10-fold symmetric “ staining dilemma configuration” can be 4-staining; 4) Through 4-staining of the four configurations of Errera, Obtained the Z-staining program (also called Theorem 4), and using this program and mathematical induction, gave the 10-fold symmetric “staining dilemma configuration” 4-staining proof. Completed the complete and concise manual proof of the four-color conjecture.