Distinguishing numbers of Cartesian products of multiple complete graphs

Abstract

We examine the distinguishing number of the Cartesian product of an arbitrary number of complete graphs. We show that for u 1 ≤ ... ≤ u d the distinguishing number of the Cartesian product of complete graphs of these sizes is either ⌈ u d 1/ s ⌉ or ⌈ u d 1/ s ⌉ + 1 where s = Π i = 1 d − 1 u i . In most cases, which of these values it is can be explicitly determined.