On Zero-Sum {mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${phi}$ from E into A ? {0} such that for some ${a in A, sum_{e in E(v)} phi(e) = a}$ for every ${v in V}$ , where E(v) is the set of edges incident to v. If ${phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${mathbb{N}}$ (the positive integers) such that ${1 in zim(G)}$ if and only if G is zero-sum ${mathbb{Z}}$ -magic and ${k geq 2 in zim(G)}$ if and only if G is zero-sum ${mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = mathbb{N} - {2}}$ or ${mathbb{N} - {2,4}.}$