Zero-Sum Magic Labelings and Null Sets of Regular Graphs

For every h ∈ 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) → Zh\{0} such that the induced vertex labeling s : V (G) → Zh, defined by s(v) = ∑ uv∈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 ∈ 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) = N and for odd r (r 6= 5), N \ {2, 4} ⊆ N(G). Moreover, we prove that if r is odd and G is a 2-edge connected r-regular graph (r 6= 5), then N(G) = N \ {2}. Also, we show that if G is a 2-edge connected bipartite graph, then N \ {2, 3, 4, 5} ⊆ N(G).