The distinguishing number of the augmented cube and hypercube powers

The distinguishing number of a graph GG, denoted D(G)D(G), is the minimum number of colors such that there exists a coloring of the vertices of GG where no nontrivial graph automorphism is color-preserving. In this paper, we answer an open question posed in Bogstad and Cowen [The distinguishing number of the hypercube, Discrete Math. 283 (2004) 29–35] by showing that the distinguishing number of Qnp, the ppth graph power of the nn-dimensional hypercube, is 2 whenever 2