2-distance coloring of planar graphs with maximum degree 5

A 2-distance coloring of a graph is a proper Formula: see text-coloring in which any two vertices with distance at most two get different colors. The 2-distance number is the smallest number Formula: see text such that Formula: see text has a 2-distance Formula: see text-coloring, denoted as Formula: see text. In 1977, Wegner conjectured that for each planar graph Formula: see text with maximum degree Formula: see text, Formula: see text if Formula: see text, Formula: see text if Formula: see text, and Formula: see text if Formula: see text. In 2001, Thomassen supported the conjecture by proving the case Formula: see text. The conjecture is still open even for Formula: see text. In this paper, we show that Formula: see text for the case Formula: see text which improves the upper bound 18 recently obtained by Hou et al.