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 distance at most 2 from each other receive different colors. Suppose that G is a planar graph with girth 5 and maximum degree \(\Delta \) $\mathrm{\Delta }$ . We prove that G admits a 2-distance \(\Delta +7\) $\mathrm{\Delta }+7$ 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 \(\Delta +6\) $\mathrm{\Delta }+6$ coloring when \(\Delta \ge 10\) $\mathrm{\Delta }\ge 10$ .