List Distinguishing Number of \(p^{\text {th}}\) Power of Hypercube and Cartesian Powers of a Graph

A graph G is said to be k-distinguishable if every vertex of the graph can be colored from a set of k colors such that no non-trivial automorphism fixes every color class. The distinguishing number D(G) is the least integer k for which G is k-distinguishable. If for each \(v\in V(G)\) we have a list L(v) of colors, and we stipulate that the color assigned to vertex v comes from its list L(v) then G is said to be \(\mathcal {L}\)-distinguishable where \(\mathcal {L}=\{L(v)\}_{v\in V(G)}\). The list distinguishing number of a graph, denoted \(D_l(G)\), is the minimum integer k such that every collection of lists \(\mathcal {L}\) with \(|L(v)|=k\) admits an \(\mathcal {L}\)-distinguishing coloring. In this paper, we prove that when a connected graph G is prime with respect to the Cartesian product then \(D_l(G^r) = D(G^r)\) for \(r \ge 3\) where \(G^r\) is the Cartesian product of the graph G taken r times. The \(p^{th}\) power of a graph (Some authors use \(G^p\) to denote the pth power of G, to avoid confusion with the notation of Cartesian power of graph G we use \(G^{[p]}\) for the pth power of G.) G is the graph \(G^{[p]}\), whose vertex set is V(G) and in which two vertices are adjacent when they have distance less than or equal to p. We determine \(D_l(Q_n^{[p]})\) for all \(n \ge 7, p \ge 1\), where \(Q_n=K_2^n\) is the hypercube of dimension n.