Distinguishing numbers for graphs and groups

Abstract

A graph G is distinguished if its vertices are labelled by a map \phi: V(G) \longrightarrow {1,2,...,k} so that no graph automorphism preserves \phi. The distinguishing number of G is the minimum number k necessary for \phi to distinguish the graph. It is one measure of the complexity of the graph. We extend these definitions to an arbitrary group action of G on a set X. A labelling \phi: X \longrightarrow {1,2,...,k} is distinguishing if no nontrivial element of G preserves \phi except those in the stabilizer of X. The distinguishing number of the group action on X is the minimum k needed for \phi to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of the symmetric group S_n on a set with distinguishing number n.