Distinguishing Cartesian powers of graphs

Abstract

Abstract:The distinguishing number D(G) of a graph is the least integer d such that there is a d-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G = K 2 , K 3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [Electron J Combin, 12 (2005), #N17] on powers of prime graphs, and results of Klavžar and Zhu [Eu J Combin, to appear]. More generally, we also prove that d(G H) = 2 if G and H are relatively prime and |H| ≤ |G| < 2 |H| − |H|. Under additional conditions similar results hold for powers of graphs with respect to the strong and the direct product.

Improving upper bounds for the distinguishing index

“…In the proof of Proposition 3.10 we also make use of an earlier result of Imrich and Klavžar [15] which is a slightly weaker version of Theorem 3.7 for d = 2.…”

“…If p = q, then p ≥ 4, and there exists a spanning asymmetric tree of K p,p (see [17]). If p < q ≤ 2 p − p + 1, then for the proof of Theorem 3.9, Imrich and Klavžar in [15] constructed a distinguishing vertex 2-colouring of K p 2K q that corresponds to a distinguishing edge 2-colouring f of K p,q , where a colouring of vertices in a K q -layer can be represented by a sequence from {1, 2} q and it corresponds to a colouring of edges incident to a vertex in P (the same is true for K p -layers and vertices in Q). We wish to show that this colouring yields a connected asymmetric subgraph of K p,q which is spanning or almost spanning.…”

“…Following [15] for p < q < 2 p − p + 1, we exclude a relevant number of such pairs of sequences of colours that the sum of them is a sequence (3, . .…”

A Note on Edge-Connectivity of the Cartesian Product of Graphs

“…The concept of Cartesian products in graph theory can be traced back to a fundamental paper of Sabidussi [8]; we note that an interesting monograph on the topic by Imrich et al [4] was published recently. In the present paper, we deal with the edgeconnectivity of the Cartesian product of two graphs.…”

Motion and distinguishing number two

“…Gluck's Theorem is a good example. If one takes a Cartesian product of enough copies of the same graph, then the distinguishing number is two (see [1,15]). For all maps with more than 10 vertices, the action of the automorphism group on the vertices has distinguishing number two (see [20]).…”