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 $ϕ$ from $E$ into $\mathbb{A} - \{0\}$ such that for some $c \in \mathbb{A}$, $\sum_{e \in E(v)} ϕ(e) = c$ 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 $ϕ$ 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 $ζ_{2j}(G)$ 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 $ζ_{2j}(G)$ when $ζ_{2j}(G)$ is finite, and show that $ζ_{2j}(G)$ is finite for all $r$-regular $G$, $r \geq 2$. Appealing to classical results on the factors of cubic graphs, we prove that $ζ_4(G) \leq 2$ 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.