#### Document Type

Article

#### Publication Date

9-16-2004

#### Publication Title

Electronic Journal of Combinatorics

#### Abstract

A graph G is distinguished if its vertices are labelled by a map φ: V(G) → {1,2,..., k} so that no non-trivial graph automorphism preserves φ. The distinguishing number of G is the minimum number k necessary for φ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of Γ on a set X. A labelling φ: X → {1, 2,..., k} is distinguishing if no element of Γ preserves Γ except those which fix each element of X. The distinguishing number of the group action on X is the minimum k needed for φ to distinguish the group action. We show that distinguishing group actions is a more general problem than distinguishing graphs. We completely characterize actions of (S)n (on a set with distinguishing number)n, answering an open question of Albertson and Collins.

#### Volume

11

#### Issue

1 R

#### First Page

1

#### Last Page

13

#### DOI

10.37236/1816

#### ISSN

10778926

#### Recommended Citation

Tymoczko, Julianna, "Distinguishing Numbers for Graphs and Groups" (2004). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.

https://scholarworks.smith.edu/mth_facpubs/112

## Comments

Peer reviewed accepted manuscript.