Face Antimagic Labelings of Plane Graphs \(P_a^b\)

Abstract

Suppose \(G\) is a finite plane graph with vertex set \(V(G)\), edge set \(E(G)\), and face set \(F(G)\). The paper deals with the problem of labeling the vertices, edges, and faces of a plane graph \(G\) in such a way that the label of a face and labels of vertices and edges surrounding that face add up to a weight of that face. A labeling of a plane graph \(G\) is called \(d\)-antimagic if for every number \(s\), the \(s\)-sided face weights form an arithmetic progression of difference \(d\). In this paper, we investigate the existence of \(d\)-antimagic labelings for a special class of plane graphs.