A note on face coloring entire weightings of plane graphs
Stanislav Jendrol′, Peter Šugerek · Discussiones Mathematicae Graph Theory · 2013
Given a weighting of all elements of a 2-connected plane graph G = (V, E, F ), let f () denote the sum of the weights of the edges and vertices incident with the face and also the weight of . Such an entire weighting is a proper face colouring provided that f () = f () for every two faces and sharing an edge. We show that for every 2-connected plane graph there is a proper face-colouring entire weighting with weights 1 through 4. For some families we improved 4 to 3.