A 2-distance (2Δ+7)-coloring of planar graphs

A vertex coloring of a graph G is called a $2$-distance coloring if any two vertices at a distance at most $2$ from each other receive different colors. Recently, Bousquet et al. (Discrete Mathematics, 346(4), 113288, 2023) proved that 2Δ+7 colors are sufficient for the $2$-distance coloring of planar graphs with maximum degree Δ≥ 9. In this paper, we strengthen their result by removing the maximum degree constraint and show that all planar graphs admit a 2-distance (2Δ+7)-coloring.This particularly improves the result of Van den Heuvel and McGuinness (Journal of Graph Theory, 42(2), 110-124, 2003).