List Distinguishing Number of 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 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 -distinguishable where . The list distinguishing number of a graph, denoted , is the minimum integer k such that every collection of lists with admits an -distinguishing coloring. In this paper, we prove that when a connected graph G is prime with respect to the Cartesian product then for where is the Cartesian product of the graph G taken r times. The power of a graph (Some authors use to denote the p th power of G , to avoid confusion with the notation of Cartesian power of graph G we use for the p th power of G .) G is the graph , 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 for all , where is the hypercube of dimension n .