On the structural result on normal plane maps

We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by ${\displaystyle 6+\left[\frac{2D+12}{D-2}\right]({(D-1)}^{(t-1)}-1)}$. This improves a recent bound ${\displaystyle 6+\left[\frac{3D+3}{D-2}\right]({(D-1)}^{t-1}-1)}$, D ≥ 8 by Jendrol' and Skupień, and the upper bound for distance-2 chromatic number.