On Friendly Index Sets of Prisms and Mébius Ladders

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $A$ be an abelian group. A labeling $f:V(G)\to A$ induces an edge labeling $f^{\ast }:E(G)\to A$ defined by $f^{\ast }(xy)=f(x)+f(y)$ for each edge $xy\in E(G)$. For $i\in A$, let $v_f(i)=|\{v\in V(G):f(v)=i\}|$, and $e_f(i)=|\{e\in E(G):f^{\ast }(e)=i\}|$. Define $c(f)=\{|e_f(i)–e_f(j)|:(i,j)\in A\times A\}$.A labeling $f$ of a graph $G$ is said to be A-friendly if $|{\text{v}}_f(i)–{\text{v}}_f(j)|\le 1$ for all $(i,j)\in A\times A$. If $c(f)$ is a $(0,1)$-matrix for an A-friendly labeling $f$, then $f$ is said to be A-cordial.When $A={\mathbb{Z}}_2$, the friendly index set of the graph $G$, denoted $FI(G)$, is defined as $\{|e_f(0)–e_f(1)|:\text{the vertex labeling }f\text{ is }{\mathbb{Z}}_2\text{-friendly}\}$.In this paper, we completely determine the friendly index sets for two classes of cubic graphs, prisms and Möbius ladders , are completely determined.