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

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 $${\mathbb{N} - \{2,4\}.}$$