Triangulation Graph and Color Changing Channel

TL;DR

This paper explores the properties of triangulation graphs and introduces a color change channel mechanism that reflects graph integrity and aids in proving structural reducibility, advancing understanding of planar graph coloring.

Contribution

It presents a novel color change channel approach for triangulation graphs and proves the reducibility of a key graph structure, negating the smallest counterexample.

Findings

Color change channel reflects triangulation graph integrity.

Proved reducibility of the structure with d(v)=5.

Negated the existence of the smallest counterexample.

Abstract

Triangulation graph staining is sufficient for planar graph staining. This article will focus on triangulation and the nature of the color change channel of the staining tool. By construction, the four colors of the vertex are converted into three colors of the side, and two colors of the triangle. Thereby the equivalent dyeing scheme is combined. The color change channel utilizes the transferability of edge dyeing to reflect the integrity of the triangulation graph. According to the nature of the color change channel, two constraints are obtained, so that the color change can be followed regularly. Eventually, a breakthrough was made in the structure of the reduced d(v)=5, which proves the reducible of this structure. This negates the existence of the smallest counterexample.