2-Distance coloring of planar graphs with maximum degree 5

A 2-distance coloring of a graph is a proper k-coloring in which any two vertices with distance at most two get different colors. The 2-distance number is the smallest number k such that G has a 2-distance k-coloring, denoted as chi(2)(G). In 1977, Wegner conjectured that for each planar graph G with maximum degree Delta, chi(2)(G) <= 7 if Delta <= 3, chi(2)(G) <= Delta + 5 if 4 <= Delta <= 7, and chi(2)(G) <= left perpendicular3 Delta/2right perpendicular + 1 if Delta >= 8. In 2001, Thomassen supported the conjecture by proving the case Delta = 3. The conjecture is still open even for Delta = 4. In this paper, we show that chi(2)(G) <= 17 for the case Delta <= 5 which improves the upper bound 18 recently obtained by Hou et al.