Geodesic
A note on vertex colorings of plane graphs
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 4, pp. 849-855

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Given an integer valued weighting of all elements of a 2-connected plane graph G with vertex set V, let c(v) denote the sum of the weight of v ∈ V and of the weights of all edges and all faces incident with v. This vertex coloring of G is proper provided that c(u) ≠ c(v) for any two adjacent vertices u and v of G. We show that for every 2-connected plane graph there is such a proper vertex coloring with weights in 1, 2, 3. In a special case, the value 3 is improved to 2.
Keywords: plane graph, vertex coloring 
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Fabrici, Igor; Jendrol’, Stanislav; Soták, Roman. A note on vertex colorings of plane graphs. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 4, pp. 849-855. http://geodesic.mathdoc.fr/item/DMGT_2014_34_4_a12/