Distinguishability of Infinite Groups and Graphs

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph Γ is said to have connectivity 1 if there exists a vertex α ∈ V Γ such that Γ \ {α} is not connected. For α ∈ V , an orbit of the point stabilizer Gα is called a suborbit of G. We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any nonnull, infinite, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertextransitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number 2. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.