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.