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We present an algorithm that determines the prime factors of connected graphs with respect to the Cartesian product in linear time and space. This improves a result of Aurenhammer et al. [Cartesian graph factorization at logarithmic cost per edge, Comput. Complexity 2 (1992) 331-349], who compute the prime factors in O(m log n) time, where m denotes the number of vertices of G and n the number of edges. Our algorithm is conceptually simpler. It gains its efficiency by the introduction of edge-labellings.

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.

The distinguishing number $D(G)$ of a graph $G$ is the least cardinal number $\aleph$ such that $G$ has a labeling with $\aleph$ labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes of infinite Cartesian products. For instance, $D(Q_{n}) = 2$, where $Q_{n}$ is the infinite hypercube of dimension ${n}$.

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