  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 $\phi$ from $E$ into $\mathbb{A} - \{0\}$ such that for
some $c \in \mathbb{A}$, $\sum_{e \in E(v)} \phi(e) = c$ 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 $\phi$ 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 $j \geq 1$,
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 $G$, $r \geq 2$.
Appealing to classical results on the factors of cubic graphs, we prove that
$\zeta_4(G) \leq 2$ 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.
