2-distance -coloring of planar graphs with girth six and

It was proved in [Z. Dvořàk, D. Kràl, P. Nejedlỳ, R. Škrekovski, Coloring squares of planar graphs with girth six, European J. Combin. 29 (4) (2008) 838–849] that every planar graph with girth g ≥ 6 and maximum degree Δ ≥ 8821 is 2-distance ( Δ + 2 ) -colorable. We prove that every planar graph with g ≥ 6 and Δ ≥ 18 is 2-distance ( Δ + 2 ) -colorable.