Entire Labeling of Plane Graphs

Abstract

A face irregular entire k-labeling ϕ : V ( E( F ! f 1, 2,..., kg of a 2-connected plane graph G= (V, E, F) is a labeling of vertices, edges and faces of G in such a way that for any two different faces f and g their weights wϕ( f) and wϕ(g) are distinct. The weight of a face f under a k-labeling ϕ is the sum of labels carried by that face and all the edges and vertices incident with the face. The minimum k for which a plane graph G has a face irregular entire k-labeling is called the entire face irregularity strength. We investigate a face irregular entire labeling as a modification of the well -known vertex irregular and edge irregular total labelings of graphs. We obtain some estimations on the entire face irregularity str ength and determine the precise values for graphs from three families of plane graphs.