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 ∆(G) ≤ D, D ≥ 12 is proved to be upper bounded by 6+ 2D+12 D−2 ((D−1)(t−1)−1). This improves a recent bound 6 + 3D+3 D−2 ((D − 1)t−1 − 1), D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.