The distinguishing number of the augmented cube and hypercube powers

The distinguishing number of a graph image, denoted image, is the minimum number of colors such that there exists a coloring of the vertices of image 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 image, the imageth graph power of the image-dimensional hypercube, is 2 whenever image. This completes the study of the distinguishing number of hypercube powers. We also compute the distinguishing number of the augmented cube image, a variant of the hypercube introduced in Choudum and Sunitha [Augmented cubes, Networks 40 (2002) 71–84]. We show that image; image; image; and image for image. The sequence of distinguishing numbers image answers a question raised in Albertson and Collins [An introduction to symmetry breaking in graphs, Graph Theory Notes N.Y. 30 (1996) 6–7].