On zero-sum ${\mathbb{Z}}_{2j}^k$-magic graphs

Abstract

Let $G=(V,E)$ be a finite graph and let $(\mathbb{A},+)$ be an abelian group with identity 0. Then $G$ is \textit{$\mathbb{A}$-magic} if and only if there exists a function $\varphi$ from $E$ into $\mathbb{A}-\{0\}$ such that for some $c\in \mathbb{A}$, ${\sum }_{e\in E(v)}\varphi (e)=c$ for every $v\in V$, where $E(v)$ is the set of edges incident to $v$. Additionally, $G$ is \textit{zero-sum $\mathbb{A}$-magic} if and only if $\varphi$ exists such that $c=0$. We consider zero-sum $\mathbb{A}$-magic labelings of graphs, with particular attention given to $\mathbb{A}={\mathbb{Z}}_{2j}^k$. For $j\ge 1$, let ${\zeta }_{2j}(G)$ be the smallest positive integer $c$ such that $G$ is zero-sum ${\mathbb{Z}}_{2j}^c$-magic if $c$ exists; infinity otherwise. We establish upper bounds on ${\zeta }_{2j}(G)$ when ${\zeta }_{2j}(G)$ is finite, and show that ${\zeta }_{2j}(G)$ is finite for all $r$-regular $G$, $r\ge 2$. Appealing to classical results on the factors of cubic graphs, we prove that ${\zeta }_4(G)\le 2$ for a cubic graph $G$, with equality if and only if $G$ has no 1-factor. We discuss the problem of classifying cubic graphs according to the collection of finite abelian groups for which they are zero-sum group-magic.