# Distinguishability of infinite groups and graphs

Smith, Simon M., Tucker, Thomas W. and Watkins, Mark E. (2012) Distinguishability of infinite groups and graphs. The Electronic Journal of Combinatorics, 19 (2). P27. ISSN 1077-8926

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## Abstract

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 Gamma is said to have 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 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 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.

Keywords: distinguishing number, primitive permutation group, infinite motion, Cartesian product of graphs G Mathematical and Computer Sciences > G110 Pure Mathematics College of Science > School of Mathematics and Physics 27493 12 May 2017 09:46

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