Face Antimagic Labelings for a Special Class of Plane Graphs $C_a^b$

Abstract

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 $\lambda :V(G)\cup E(G)\cup F(G)\to \{1,2,3,\dots ,|V(G)|+|E(G)|+|F(G)|\}$ 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 $d$-$antimagic$ if for every number $s\ge 3$, the set of $s$-sided face weights is$W_s=\{a_s+id:0\le i\le f_s\}$ for some integers $a_s$ and $d$ ($a>0$, $d\ge 0$), 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 $d$-$antimagic$ 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 $d\in \{a-2,a-1,a+1,a+2\}$.