Labelings of Plane Graphs with Determined Face Weights

This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examine the existence of such labelings for several families of plane graphs.