Distinguishing Cartesian Powers of Graphs

Given a graph G, a labeling c : V (G) → {1, 2, . . . , d} is said to be d-distinguishing if the only element in Aut(G) that preserves the labels is the identity. The distinguishing number of G, denoted by D(G), is the minimum d such that G has a d-distinguishing labeling. If G2H denotes the Cartesian product of G and H, let G 2 = G2G and G r = G2G r−1 . A graph G is said to be prime with respect to the Cartesian product if whenever G ∼= G12G2, then either G1 or G2 is a singleton vertex. This paper proves that if G is a connected, prime graph, then D(G r ) = 2 whenever r ≥ 4.