2-Distance coloring of planar graphs with girth 5

A vertex coloring is said to be 2-distance if any two distinct vertices of distance at most 2 receive different colors. Let G be a planar graph with girth at least 5. In this paper, we prove that G admits a 2-distance coloring with at most Δ(G)+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (G)+3$$\end{document} colors if Δ(G)≥339\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (G)\ge 339$$\end{document}.