Color identical pairs in 4-chromatic graphs

TL;DR

This paper explores a property of 4-chromatic graphs related to color identical pairs and their implications for planar supergraphs, providing insights connected to the Four Color Theorem.

Contribution

It establishes a new equivalence between color identical pairs in 4-chromatic graphs and the absence of certain planar supergraphs, offering a novel perspective on the Four Color Theorem.

Findings

Color identical pairs imply no planar supergraph with adjacent u and v.

The property characterizes 4-chromatic graphs related to the Four Color Theorem.

Provides a new criterion for understanding graph colorings and planarity.

Abstract

I argue that, given vertices u and v in a 4-chromatic graph G, if the color of u equals the color of v in every 4-coloring of G then G has no planar supergraph where u and v are adjacent. This is equivalent to the Four Color Theorem.