The distinguishing number of Cartesian products of complete graphs

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d labels that is preserved only by a trivial automorphism. We prove that Cartesian products of relatively prime graphs whose sizes do not differ too much can be distinguished with a small number of colors. We determine the distinguishing number of the Cartesian product Kk Kn for all k and n, either explicitly or by a short recursion. We also introduce column-invariant sets of vectors and prove a switching lemma that plays a key role in the proofs. Key words: Distinguishing number; Automorphism; Cartesian product; Complete graphs AMS subject classification (2000): 05C25 This work was supported in part by the Ministry of Science of Slovenia under the grants P1-0297 and J1-6150. 1 

• Type: Journal
• Year: 2008
• Where: EJC
• Authors: Wilfried Imrich, Janja Jerebic, Sandi Klavzar
• Keywords: Cartesian Product, Cartesian Product Kk, EJC 2007, EJC 2008, Information Technology, Trivial Automorphism