On zero-sum ℤ_2j^k -magic graphs

Let G = (V,E) be a finite graph and let (𝔸,+) be an abelian group with identity 0. Then G is 𝔸 - magic if and only if there exists a function ϕ from E into 𝔸 - {0} such that for some c ∈𝔸, ∑ _e ∈ E(v)ϕ (e) = c for every v ∈ V , where E ( v ) is the set of edges incident to v . Additionally, G is zero-sum 𝔸 - magic if and only if ϕ exists such that c = 0 . We consider zero-sum 𝔸 -magic labelings of graphs, with particular attention given to 𝔸 = ℤ_2j^k . For j ≥ 1 , let ζ _2j(G) be the smallest positive integer c such that G is zero-sum 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 ≥ 2 . Appealing to classical results on the factors of cubic graphs, we prove that 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.