On Group-Magic Trees, Double Trees and Abbreviated Double Trees

Abstract

For $k>0$, we call a graph $G=(V,E)$ $k$-magic if there exists a labeling $l:E(G)\to {\mathbb{Z}}_k^{\ast }$ such that the induced vertex set labeling $l^{+}:V(G)\to {\mathbb{Z}}_k$, defined by

$$l^{+}(v)=\sum \{l(u,v):(u,v)\in E(G)\}$$

is a constant map. We denote the set of all $k$ such that $G$ is $k$-magic by $\text{IM}(G)$. We call this set the \textbf{\emph{integer-magic spectrum}} of $G$. We investigate these sets for trees, double trees, and abbreviated double trees. We define group-magic spectrum for $G$ similarly. Finally, we show that a tree is $k$-magic, $k>2$, if and only if it is $k$-label reducible.