A renewal approach to prove the Four Color Theorem unplugged, Part I: RGB-tilings on maximal planar graphs

TL;DR

This paper introduces a novel renewal approach using RGB-tilings on maximal planar graphs as a foundation for proving the Four Color Theorem without computer assistance, setting the stage for subsequent parts.

Contribution

It develops the concept of R/G/B-tilings and their properties on maximal planar graphs, proposing a new framework for the Four Color Theorem proof.

Findings

Defined R/G/B-tilings and RGB-tilings on MPG's

Established properties of tilings and canal lines

Introduced the concept of canal systems in graph coloring

Abstract

This is the first part of three episodes to demonstrate a renewal approach for proving the Four Color Theorem without checking by a computer. The second and the third episodes have subtitles: ``R/G/B Kempe chains in an extremum non-4-colorable MPG'' 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. In this first part, we introduce R/G/B-tilings as well as their tri-coexisting version RGB-tiling on an MPG or a semi-MPG. We associate these four kinds of edge-colorings with 4-colorings by 1/2/3/4 on vertices in MPS's or semi-MPG's. Several basic properties for tilings on MPG's and semi-MPG's are developed. Especially the idea of R/G/B-canal lines, as well as canal system, is a cornerstone. This work started on May 31, 2018 and was first announced by the author~\cite{Liu2020} at the Institute of Mathematics, Academia Sinica, Taipei, Taiwan, on Jan.\ 22, 2020, when the pandemic just occurred.