List-Distinguishing Cartesian Products of Cliques

The distinguishing number of a graph $G$, denoted $D(G)$, is the minimum number of colors needed to produce a coloring of the vertices of $G$ so that every nontrivial isomorphism interchanges vertices of different colors. A list assignment $L$ on a graph $G$ is a function that assigns each vertex of $G$ a set of colors. An $L$-coloring of $G$ is a coloring in which each vertex is colored with a color from $L(v)$. The list distinguishing number of $G$, denoted $D_{\ell}(G)$ is the minimum $k$ such that every list assignment $L$ that assigns a list of size at least $k$ to every vertex permits a distinguishing $L$-coloring. In this paper, we prove that when and $n$ is large enough, the distinguishing and list-distinguishing numbers of $K_n\Box K_m$ agree for almost all $m>n$, and otherwise differ by at most one. As a part of our proof, we give (to our knowledge) the first application of the Combinatorial Nullstellensatz to the graph distinguishing problem and also prove an inequality for the binomial distribution that may be of independent interest.