The distinguishing number of the augmented cube and hypercube powers

Abstract

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 Bogstad and Cowen [The distinguishing number of the hypercube, Discrete Math. 283 (2004) 29–35] by showing that the distinguishing number of Q n p , 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 AQ n , a variant of the hypercube introduced in Choudum and Sunitha [Augmented cubes, Networks 40 (2002) 71–84]. We show that D ( AQ 1 ) = 2 ; D ( AQ 2 ) = 4 ; D ( AQ 3 ) = 3 ; and D ( AQ n ) = 2 for n ⩾ 4 . The sequence of distinguishing numbers { D ( AQ n ) } n = 1 ∞ answers a question raised in Albertson and Collins [An introduction to symmetry breaking in graphs, Graph Theory Notes N.Y. 30 (1996) 6–7].