On Constructions which yield Fully Magic Graphs

Abstract

For any abelian group \(A\), we denote \(A^*=A-\{0\}\). Any mapping \(1: E(G) \to A^*\) is called a labeling. Given a labeling on the edge set of \(G\) we can induce a vertex set labeling \(1^+: V(G) \to A\) as follows:

\[1^+(v) = \Sigma\{1(u,v): (u,v) \in E(G)\}.\]

A graph \(G\) is known as \(A\)-magic if there is a labeling \(1: E(G) \to A^*\) such that for each vertex \(v\), the sum of the labels of the edges incident to \(v\) are all equal to the same constant; i.e., \(1^+(v) = c\) for some fixed \(c\) in \(A\). We will call \(\langle G,\lambda \rangle\) an \(A\)-magic graph with sum \(c\).

We call a graph \(G\) fully magic if it is \(A\)-magic for all non-trivial abelian groups \(A\). Low and Lee showed in \([11]\) if \(G\) is an eulerian graph of even size, then \(G\) is fully magic. We consider several constructions that produce infinite families of fully magic graphs. We show here every graph is an induced subgraph of a fully magic graph.