A Note on 3-Distance Coloring of Planar Graphs

In 2018, Thomassen showed that every subcubic planar graph has the $2$-distance chromatic number at most $7$, which was originally conjectured by Wegner (1977). In this note, we consider $3$-distance colorings of this family of graphs, and prove that every subcubic planar graph has $3$-distance chromatic number at most $17$, and we conjecture that this number can be reduced to $12$. In addition, we show that every planar graph $G$ with maximum degree at most $\Delta$ has $3$-distance chromatic number at most $(6+o(1))\Delta$.