A Result on the 2-Distance Coloring of Planar Graphs with Girth Five

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 of $G$, and denoted by $\chi_2(G)$. We prove that if $G$ is a planar graph with girth $5$ and maximum degree $\Delta \geq 12$, then $\chi_2(G)\leq \Delta(G)+5$.