Magic labelings of regular graphs

Abstract

Let G(V, E) be a graph and λ be a bijection from the set V ≼ E to the set of the first |V| + |E| natural numbers. The weight of a vertex is the sum of its label and the labels of all adjacent edges. We say λ is a vertex magic total (VMT) labeling of G if the weight of each vertex is constant. We say λ is an (s, d)-vertex antimagic total (VAT) labeling if the vertex weights form an arithmetic progression starting at s with difference d.J. MacDougall conjectured that any regular graph with the exception of K2 and 2K3 has a VMT labeling. We give constructions of VAT labelings of any even-regular graphs and VMT labelings of certain regular graphs.