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

Let $$G = (V,E)$$ G = ( V , E ) be a finite graph and let $$(\mathbb {A},+)$$ ( A , + ) be an abelian group with identity 0. Then G is $$\mathbb {A}$$ A -magic if and only if there exists a function $$\phi $$ ϕ from E into $$\mathbb {A} - \{0\}$$ A - { 0 } such that for some $$c \in \mathbb {A}, \sum _{e \in E(v)} \phi (e) = c$$ c ∈ A , ∑ e ∈ E ( v ) ϕ ( e ) = c for every $$v \in V$$ v ∈ V , where E(v) is the set of edges incident to v. Additionally, G is zero-sum $$\mathbb {A}$$ A -magic if and only if $$\phi $$ ϕ exists such that $$c = 0$$ c = 0 . We consider zero-sum $$\mathbb {A}$$ A -magic labelings of graphs, with particular attention given to $$\mathbb {A} = \mathbb {Z}_{2j}^k$$ A = Z 2 j k . For $$j \ge 1$$ j ≥ 1 , let $$\zeta _{2j}(G)$$ ζ 2 j ( G ) be the smallest positive integer c such that G is zero-sum $$\mathbb {Z}_{2j}^c$$ Z 2 j c -magic if c exists; infinity otherwise. We establish upper bounds on $$\zeta _{2j}(G)$$ ζ 2 j ( G ) when $$\zeta _{2j}(G)$$ ζ 2 j ( G ) is finite, and show that $$\zeta _{2j}(G)$$ ζ 2 j ( G ) is finite for all r-regular $$G, r \ge 2$$ G , r ≥ 2 . Appealing to classical results on the factors of cubic graphs, we prove that $$\zeta _4(G) \le 2$$ ζ 4 ( G ) ≤ 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.