Face Antimagic Labelings for a Special Class of Plane Graphs \(C_{a}^{b}\)

Suppose \( G = (V,E,F) \) is a finite plane graph with vertex set \( V(G) \), edge set \( E(G) \), and face set \( F(G) \). A bijection \( \lambda: V(G) \cup E(G) \cup F(G) \rightarrow \{1,2,3,\ldots,|V(G)| + |E(G)| + |F(G)|\} \) is called a labeling of type \( (1,1,1) \). The weight of a face under a labeling is the sum of the labels (if present) carried by that face and the edges and vertices surrounding it. A labeling of a plane graph \( G \) is called \( d \)-\emph{antimagic} if for every number \( s \geq 3 \), the set of \( s \)-sided face weights is

\[
W_s = \{a_s + id: 0 \leq i \leq f_s\}
\]

for some integers \( a_s \) and \( d \) (\( a > 0 \), \( d \geq 0 \)), where \( f_s \) is the number of \( s \)-sided faces. We allow different sets \( W_s \) for different \( s \).

In this paper, we deal with \( d \)-\emph{antimagic} labelings of type \( (1,1,1) \) for a special class of plane graphs \( C_a^b \) and we show that a \( C_a^b \) graph has \( d \)-antimagic labeling for \( d \in \{a-2,a-1,a+1,a+2\} \).