Some results on 2-distance coloring of planar graphs with girth five

TL;DR

This paper improves bounds on the number of colors needed for 2-distance coloring of planar graphs with girth five, providing tighter results for graphs with higher maximum degree.

Contribution

It establishes new upper bounds for 2-distance coloring of planar graphs with girth five, improving previous results by Dong and Lin.

Findings

2-distance $ riangle+7$ coloring for all such graphs

2-distance $ riangle+6$ coloring when $ riangle extgreater= 10$

Enhanced understanding of coloring bounds for planar graphs with girth five

Abstract

A vertex coloring of a graph $G$ is called a 2-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Suppose that $G$ is a planar graph with girth $5$ and maximum degree $\Delta$. We prove that $G$ admits a $2$-distance $\Delta+7$ coloring, which improves the result of Dong and Lin (J. Comb. Optim. 32(2), 645-655, 2016). Moreover, we prove that $G$ admits a $2$-distance $\Delta+6$ coloring when $\Delta\geq 10$.