Distinguishing Cartesian Powers of Graphs

Given a graph $G$, a labeling $c:V(G) \rightarrow \{1, 2, \ldots, d\}$ is said to be $d$-distinguishing if the only element in ${\rm Aut}(G)$ that preserves the labels is the identity. The distinguishing number of $G$, denoted by $D(G)$, is the minimum $d$ such that $G$ has a $d$-distinguishing labeling. If $G \square H$ denotes the Cartesian product of $G$ and $H$, let $G^{^2} = G \square G$ and $G^{^r} = G \square G^{^{r-1}}$. A graph $G$ is said to be prime with respect to the Cartesian product if whenever $G \cong G_1 \square G_2$, then either $G_1$ or $G_2$ is a singleton vertex. This paper proves that if $G$ is a connected, prime graph, then $D(G^{^r}) = 2$ whenever $r \geq 4$.