On \(d\)-Antimagic Labelings for a Special Class of Plane Graphs

Abstract

A bijection \( \lambda: V \cup E \cup F \to \{1, 2, 3, \dots, |V| + |E| + |F|\} \) is called a \( d \)-antimagic labeling of type \( (1, 1, 1) \) of plane graph \( G(V, E, F) \) if the set of \( s \)-sided face weights is \( W_s = \{a_s + a_s+d, a_s+2d, \dots, a_s + (f_s-1)d\} \) for some integers \( s \), \( a_s \), and \( d \), where \( f_s \) is the number of \( s \)-sided faces and the face weight is the sum of the labels carried by that face and the edges and vertices surrounding it. In this paper, we examine the existence of \( d \)-antimagic labelings of type \( (1, 1, 1) \) for a special class of plane graphs \( {C}_a^b \).