Local structures in plane maps and distance colourings

Abstract A structural result on normal plane maps is presented. It strengthens a result by Borodin which is related to Kotzig's and Lebesgue's classical results. Then the t-distance chromatic number of a planar graph G with maximum degree Δ ( G )⩽ D where D ⩾8 is proved to be bounded above by a polynomial in D of degree t −1.