Distinguishability of infinite groups and graphs

Abstract

The {\em 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 {\em distinguishing number} of a graph is the distinguishing number of its full automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have {\em connectivity 1} if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha \}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_{\alpha }$ is called a {\em suborbit} of $G$. We prove that every connected primitive graph with infinite diameter and countably many vertices has distinguishing number 2. Consequently, any infinite, connected, primitive, locally finite graph is 2-distinguishable; so, too, is any infinite primitive group with finite suborbits. We also show that all denumerable vertex-transitive 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.