2-Distance coloring of planar graphs without adjacent 5-cycles

The k-2-distance coloring of graph G is a mapping c:V(G)→{1,2,…,k}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c:V(G)\rightarrow \{1,2,\ldots ,k\}$$\end{document} such that any two vertices at distance at most two from each other get different colors. The 2-distance chromatic number is the smallest integer k such that G has a k-2-distance coloring, denoted by χ2(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{2}(G)$$\end{document}. In this paper, we prove that every planar graph G without adjacent 5-cycles and g(G)≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(G)\ge 5$$\end{document} and Δ(G)≥17\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta (G)\ge 17$$\end{document} has χ2(G)≤Δ+3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _{2}(G)\le \Delta +3$$\end{document}.