An improved bound for 2-distance coloring of planar graphs with girth six

TL;DR

This paper establishes a new upper bound of +4 for the 2-distance chromatic number of planar graphs with girth six and maximum degree at least six, improving previous bounds.

Contribution

The paper presents an improved bound for the 2-distance coloring of planar graphs with girth six, advancing the understanding of coloring properties in such graphs.

Findings

+4 upper bound for   2-distance chromatic number

Improved previous bounds for planar graphs with girth six

Applicable for graphs with maximum degree   6 or more

Abstract

A vertex coloring of a graph $G$ is said to be a 2-distance coloring if any two vertices at distance at most $2$ from each other receive different colors, and the least number of colors for which $G$ admits a $2$-distance coloring is known as the $2$-distance chromatic number $\chi_2(G)$ of $G$. When $G$ is a planar graph with girth at least $6$ and maximum degree $\Delta \geq 6$, we prove that $\chi_2(G)\leq \Delta +4$. This improves the best-known bound for 2-distance coloring of planar graphs with girth six.