Relaxation of Wegner's Planar Graph Conjecture for maximum degree 4

TL;DR

This paper relaxes the coloring constraints for the square of planar graphs with maximum degree 4, showing it can be colored with 9 colors under a weaker condition than proper coloring.

Contribution

It introduces a new relaxed coloring approach for the square of planar graphs with degree 4, reducing the number of colors needed from the conjectured 9 to 9 with a relaxed condition.

Findings

Achieves a 9-coloring under relaxed conditions for degree 4 graphs

Provides a different approach from existing conjectures and bounds

Lays groundwork for further relaxation and coloring bounds

Abstract

The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the chromatic number of the square $G^2$ of a planar graph $G$, depending on the maximum degree $\Delta(G)$ of $G$. The only case that the conjecture is resolved is when $\Delta(G)=3$, which was proven to be true by Thomassen, and independently by Hartke, Jahanbekam, and Thomas. For $\Delta(G)=4$, Wegner's Planar Graph Conjecture states that the chromatic number of $G^2$ is at most 9; even this case is still widely open, and very recently Bousquet, de Meyer, Deschamps, and Pierron claimed an upper bound of 12. We take a completely different approach, and show that a relaxation of properly coloring the square of a planar graph $G$ with $\Delta(G)=4$ can be achieved with 9 colors. Instead of requiring every color in the neighborhood of a vertex to be unique, which is equivalent to a proper coloring of $G^2$, we seek a proper coloring of $G$ such that at most one color is allowed to be repeated in the neighborhood of a vertex of degree 4, but nowhere else.