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.