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 u₁ ≤ ... ≤ 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.