A renewal approach to prove the Four Color Theorem unplugged, Part II: R/G/B Kempe chains in an extremum non-4-colorable MPG

TL;DR

This paper introduces a renewal approach using R/G/B Kempe chains to analyze extremum non-4-colorable maximal planar graphs, aiming to provide a computer-free proof of the Four Color Theorem.

Contribution

It presents a novel renewal perspective with R/G/B Kempe chains on extremum non-4-colorable graphs, enhancing understanding of 4-colorability without computer assistance.

Findings

Fundamental theorems related to R-/RGB-tilings and 4-colorability.

New if-and-only-if conditions for $EP-ackslash\{e\}$ using Kempe chains.

Insights into vertex and edge colorings through RGB-tilings.

Abstract

This is the second part of three episodes to demonstrate a renewal approach for proving the Four Color Theorem without checking by a computer. The first and the third episodes have subtitles: ``RGB-tilings on maximal planar graphs'' and ``Diamond routes, canal lines and $\Sigma$-adjustments,'' 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 second part, we refresh the false proof on $EP$ by Kempe for the Four Color Theorem. And then using single color tilings or RGB-tilings on $EP$, we offer a renewal point of view through R/G/B Kempe chains to enhance our coloring skill, either in vertex-colorings or in edge-colorings. We discover many fundamental theorems associated with R-/RGB-tilings and 4-colorability; an adventure study on One Piece, which is either an MPG or an $n$-semi-MPG; many if-and-only-if statements for $EP-\{e\}$ by using Type A or Type B $e$-diamond and Kempe chains. This work started on May 31, 2018 and was first announced by the author~\cite{Liu2020} on Jan.\ 22, 2020, when the pandemic just occurred.