Color fixation and color identity in 4-chromatic graphs

TL;DR

This paper presents a novel argument related to the Four Color Theorem, demonstrating that certain color-identical vertex pairs in 4-chromatic planar graphs cannot be joined without losing planarity.

Contribution

It introduces a new variation of an existing argument to establish a property of color-identical vertices in 4-chromatic planar graphs, providing insight into the structure of such graphs.

Findings

No 4-chromatic planar graph has a joinable pair of color identical vertices.

The result is equivalent to the Four Color Theorem.

The argument offers a new perspective on graph coloring constraints.

Abstract

I argue that there is no 4-chromatic planar graph with a joinable pair of color identical vertices, i.e., given a 4-chromatic planar graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring of G, then there is no planar supergraph of G where u and v are adjacent. This is equivalent to the Four Color Theorem. (My argument is a variation of my argument in arXiv:1402.7368)