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A Mathematical Proof of the Four-Color Conjecture (1): Transformation Step

Jin Xu

Abstract

The four-color conjecture has puzzled mathematicians for over 170 years and has yet to be proven by purely mathematical methods. This series of articles provides a purely mathematical proof of the four-color conjecture, consisting of two parts: the transformation step and the decycle step. The transformation step uses two innovative tools, contracting and extending operations and unchanged bichromatic cycles, to transform the proof of the four-color conjecture into the decycle problem of 4-base modules. Moreover, the decycle step solves the decycle problem of 4-base modules using two other innovative tools: the color-connected potential and the pocket operations. This article presents the proof of the transformation step.

A Mathematical Proof of the Four-Color Conjecture (1): Transformation Step

Abstract

The four-color conjecture has puzzled mathematicians for over 170 years and has yet to be proven by purely mathematical methods. This series of articles provides a purely mathematical proof of the four-color conjecture, consisting of two parts: the transformation step and the decycle step. The transformation step uses two innovative tools, contracting and extending operations and unchanged bichromatic cycles, to transform the proof of the four-color conjecture into the decycle problem of 4-base modules. Moreover, the decycle step solves the decycle problem of 4-base modules using two other innovative tools: the color-connected potential and the pocket operations. This article presents the proof of the transformation step.
Paper Structure (14 sections, 8 theorems, 3 equations, 21 figures)

This paper contains 14 sections, 8 theorems, 3 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Structure of MPGs
  3. Construction of MPGs
  4. Colorings of MPGs
  5. Preliminaries
  6. Unchanged Bichromatic-Cycle Maximal Planar Graphs
  7. Definitions and properties
  8. Types
  9. Base Modules
  10. Definitions and Types
  11. Properties of 4-base modules
  12. Contracting and Extending System
  13. Transformation from FCC to The Decycle Problem
  14. Basic theories of decycle colorings

Key Result

theorem 1

Let $G$ be a 4-chromatic MPG with $\delta(G)\geq 4$, $f\in C_4^0(G)$, and $C\in C^2(f)$. Then, $C$ is a UB-cycle of $f$ if and only if for any $C'\in C^2(F^f)$ with $f(C')\neq f(C)$, $C$ and $C'$ are nonintersecting.

Figures (21)

  • Figure 1: The three operators proposed by Eberhard
  • Figure 2: Edge-flipping
  • Figure 3: The three operators $\phi_4, \phi_5$ and $\phi_6$ proposed by Barnette and Butler
  • Figure 4: 55-configuration (left) and 56-configuration (right)
  • Figure 5: The UBCMPG with the minimum order
  • ...and 16 more figures

Theorems & Definitions (13)

  • theorem 1
  • proof
  • corollary 1
  • theorem 2
  • proof
  • theorem 3
  • proof
  • corollary 2
  • theorem 4
  • proof
  • ...and 3 more