Complete Multipartite Graphs and their Null Set

For every natural number h, a graph G is said to be h-magic if there exists a labelling l : E ( G ) → Z h \ { 0 } such that the induced vertex set labelling l + : V ( G ) → Z h defined by l + ( v ) = ∑ u v ∈ E ( G ) l ( u v ) , is a constant map. When this constant is zero, it is said that G admits a zero-sum h-magic labelling. The null set of a graph G, denoted by N(G), is the set of all natural numbers h ∈ N such that G admits an h-zero-sum magic labelling. In 2007, E. Salehi determined the null set of complete bipartite graphs. In this paper we generalize this result by obtaining the null set of complete multipartite graphs.