Abstract
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 \({\mathbb{N} - \{2,4\}.}\)
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Choi, JO., Georges, J.P. & Mauro, D. On Zero-Sum \({\mathbb{Z}_k}\) -Magic Labelings of 3-Regular Graphs. Graphs and Combinatorics 29, 387–398 (2013). https://doi.org/10.1007/s00373-012-1142-6
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DOI: https://doi.org/10.1007/s00373-012-1142-6