Distinguishing infinite graphs

Abstract

The distinguishing number D(G) of a graph G is the least cardinal number א such that G has a labeling with א 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(Qn) = 2, where Qn is the infinite hypercube of dimension n.