Distinguishing Cartesian Powers of Graphs Given a graph G, a labeling c:V(G) rightarrow 1, 2, ldots, d is said to be  ddistinguishing  if the only element in rm 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 ddistinguishing labeling. If G square H denotes the Cartesian product of G and H, let G2  G square G and Gr  G square Gr1. A graph G is said to be  prime  with respect to the Cartesian product if whenever G cong G_1 square G_2, then either G_1 or G_2 is a singleton vertex.  This paper proves that if G is a connected, prime graph, then D(Gr)  2 whenever r geq 4.

The EdgeCount Criterion for Graphic Lists We give a new short proof of Korenu0027s characterization of graphic lists, extended to multigraphs with bounded multiplicity p, called  pgraphs . The EdgeCount Criterion (ECC) for an integer ntuple d and integer p is the statement that for all disjoint sets I and J of indices, sum_iin I  d_isum_jin J p(n1)d_jge pI,J.  An integer list d is the degree list of a pgraph if and only if it has even sum and satisfies ECC. Analogous statements hold for bipartite or directed graphs, and an old characterization of degree lists of signed graphs follows as a corollary  of the extension to multigraphs.