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

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 $\mathrm{\Delta }$. We prove that $G$ admits a $2$-distance $\mathrm{\Delta }+7$ 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 $\mathrm{\Delta }+6$ coloring when $\mathrm{\Delta }\ge 10$.