A renewal approach to prove the Four Color Theorem unplugged, Part III: Diamond routes, canal lines and $\Sigma$-adjustments

TL;DR

This paper presents a novel, computer-free proof of the Four Color Theorem using a renewal approach with new tools like diamond routes and canal lines, focusing on extremum non-4-colorable maximal planar graphs.

Contribution

It introduces three innovative tools based on RGB-tilings and proves that no four degree-5 vertices form a diamond in extremum non-4-colorable graphs, advancing the renewal proof approach.

Findings

No four degree-5 vertices form a diamond in extremum $EP$.

Introduces diamond routes, canal lines, and $\\Sigma$-adjustments as new tools.

Supports a computer-free proof of the Four Color Theorem.

Abstract

This is the last part of three episodes to demonstrate a renewal approach for proving the Four Color Theorem without checking by a computer. The first and the second episodes have subtitles: ``RGB-tilings on maximal planar graphs'' and ``R/G/B Kempe chains in an extremum non-4-colorable MPG,'' where R/G/B stand for red, green and blue colors to paint on edges and an MPG stands for a maximal planar graph. We focus on an extremum non-4-colorable MPG $EP$ in the whole paper. In this part we introduce three tools based on RGB-tilings. They are diamond routes, normal and generalized canal lines or rings and $\Sigma$-adjustments. Using these tools, we show a major result of this paper: no four vertices of degree 5 form a diamond in any extremum $EP$.