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

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 ${\chi }_2\left(G\right)$ of $G$. When $G$ is a planar graph with girth at least 6 and maximum degree $\Delta \ge 6$, we prove that ${\chi }_2\left(G\right)\le \Delta +4$. This improves the best known bound for 2-distance coloring of planar graphs with girth six.