The distinguishing number of the augmented cube and hypercube powers

The distinguishing number of a graph G, denoted D(G), is the minimum number of colors such that there exists a coloring of the vertices of G where no nontrivial graph automorphism is color-preserving. In this paper, we answer an open question posed in [3] by showing that the distinguishing number of Qpn, the p th graph power of the n-dimensional hypercube, is 2 whenever 2 < p < n − 1. This completes the study of the distinguishing number of hypercube powers. We also compute the distinguishing number of the augmented cube AQn, a variant of the hypercube introduced in [7]. We show that D(AQ1) = 2; D(AQ2) = 4; D(AQ3) = 3; and D(AQn) = 2 for n ≥ 4. The sequence of distinguishing numbers {D(AQn)} ∞ n=1 answers a question raised in [1].