Distinguishing numbers of Cartesian products of multiple complete graphs - Fisher, Michael J. (Ars mathematica contemporanea, 2012)
We examine the distinguishing number of the Cartesian product of an arbitrary number of complete graphs. We show that for ?$u_1 \le … \le u_d$? the distinguishing number of the Cartesian product of complete graphs of these sizes is either ?$\lceil u_d^{1/s} \rceil$? or ?$\lceil u_d^{1/s} \rceil + 1$? where ?$s = \prod_{i=1}^{d-1} u_i$?. In most cases, which of these values it is can be explicitly determined.
Keywords: Cartesian product; complete graph; distinguishing number; graph theory; kartezični produkt; poln graf; razlikovalno število; teorija grafov