On Constructions which yield Fully Magic Graphs

Abstract

For any abelian group $A$, we denote $A^{\ast }=A-\{0\}$. Any mapping $1:E(G)\to A^{\ast }$ 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:

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

A graph $G$ is known as $A$-magic if there is a labeling $1:E(G)\to A^{\ast }$ 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 $⟨G,\lambda ⟩$ 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.