Geodesic
On the structural result on normal plane maps
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 2, pp. 293-303

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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 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.
Keywords: plane map, distance colouring 
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Madaras, Tomás; Marcinová, Andrea. On the structural result on normal plane maps. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 2, pp. 293-303. http://geodesic.mathdoc.fr/item/DMGT_2002_22_2_a6/