2-distance (Δ+2)(Δ+2)-coloring of planar graphs with girth six and Δ≥18Δ≥18

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≥6g≥6 and maximum degree Δ≥8821Δ≥8821 is 2-distance (Δ+2)(Δ+2)-colorable. We prove that every planar graph with g≥6g≥6 and Δ≥18Δ≥18 is 2-distance (Δ+2)(Δ+2)-colorable.