On 2-Distance ($\mathrm{\Delta }+4$)-coloring of planar graphs with girth at least five

A vertex coloring of a graph $G$ is called a 2-distance coloring if any two vertices at distance at most $2$ from each other receive different colors. Let $G$ be a planar graph with girth at least $5$. We prove that $G$ admits a $2$-distance coloring with $\mathrm{\Delta }+4$ colors if $\mathrm{\Delta }\ge 22$.