A 2-distance $(2\Delta+7)$-coloring of planar graphs

Abstract

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 $2\Delta+7$ colors are sufficient for the $2$-distance coloring of planar graphs with maximum degree $\Delta\geq 9$. In this paper, we strengthen their result by removing the maximum degree constraint and show that all planar graphs admit a 2-distance $(2\Delta+7)$-coloring. This particularly improves the result of Van den Heuvel and McGuinness (Journal of Graph Theory, 42(2), 110-124, 2003).