A note on vertex colorings of plane graphs
Igor Fabrici, Jochen Harant, Stanislav Jendrol′, Roman Soták · Discussiones Mathematicae Graph Theory · 2014
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.