The number of distinguishing colorings of a Cartesian product graph

Abstract

A vertex coloring is called distinguishing if the identity is the only automorphism that can preserve it. The distinguishing threshold θ(G) of a graph G is the minimum number of colors k required that any arbitrary k-coloring of G is distinguishing. In this paper, we calculate the  distinguishing threshold of a Cartesian product graph. Moreover, we calculate the number of non-equivalent distinguishing colorings of  grids.