A Note on 3-Distance Coloring of Planar Graphs

Thomassen (J. Combin. Theory Ser B 128:192–218, 2018) showed that every subcubic planar graph has 2-distance chromatic number at most 7, which was originally conjectured by Wegner (graphs with given diameter and a coloring problem, University of Dortmund, preprint, 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 with maximum degree at most \(\Delta \) $\mathrm{\Delta }$ has 3-distance chromatic number at most \((6+o(1))\Delta \) $(6+o(1))\mathrm{\Delta }$ .