Zero-sum magic labelings and null sets of regular graphs (Q405209)

Summary: For every \(h\in \mathbb{N}\), a graph \(G\) with the vertex set \(V(G)\) and the edge set \(E(G)\) is said to be \(h\)-magic if there exists a labeling \(l : E(G) \to\mathbb{Z}_h \setminus \{0\}\) such that the induced vertex labeling \(s : V (G) \to \mathbb{Z}_h\), defined by \(s(v) =\sum_{uv \in E(G)} l(uv)\) is a constant map. When this constant is zero, we say that \(G\) admits a zero-sum \(h\)-magic labeling. The null set of a graph \(G\), denoted by \(N(G)\), is the set of all natural numbers \(h \in \mathbb{ N} \) such that \(G\) admits a zero-sum \(h\)-magic labeling. In 2012, the null sets of 3-regular graphs were determined. In this paper we show that if \(G\) is an \(r\)-regular graph, then for even \(r\) (\(r > 2\)), \(N(G)=\mathbb{N}\) and for odd \(r\) (\(r\neq5\)), \(\mathbb{N} \setminus \{2,4\}\subseteq N(G)\). Moreover, we prove that if \(r\) is odd and \(G\) is a \(2\)-edge connected \(r\)-regular graph (\(r\neq 5\)), then \( N(G)=\mathbb{N} \setminus \{2\}\). Also, we show that if \(G\) is a \(2\)-edge connected bipartite graph, then \(\mathbb{N} \setminus \{2,3,4,5\}\subseteq N(G)\).