Abstract

The distinguishing number D(G) of a graph G is the least integer d such that there is a d-labeling of the vertices of G which is not preserved by any nontrivial automorphism. For a graph G let G^r be the rth power of G with respect to the Cartesian product. It is proved that D(G^r)=2 for any connected graph G with at least 3 vertices and for any r≥3. This confirms and strengthens a conjecture of Albertson. Other graph products are also considered and a refinement of the Russell and Sundaram motion lemma is proved.

Mathematical subject codes: 05C25

Article Outline

1. Introduction

2. An upper bound on D(G*H)

3. A refinement of the motion lemma and the main result

Acknowledgements

References