Choice-Distinguishing Colorings of Cartesian Products of Graphs Tomlins, Christian James Applied mathematics Mathematics Choice number Choice-distinguishing number Distinguishing number Graph theory List coloring Symmetry breaking A coloring $f: V(G)\rightarrow \mathbb N$ of a graph $G$ is said to be \emph{distinguishing} if no non-identity automorphism preserves every vertex color. The distinguishing number, $D(G)$, of a graph $G$ is the smallest positive integer $k$ such that there exists a distinguishing coloring $f: V(G)\rightarrow [k]$ and was introduced by Albertson and Collins in their paper ``Symmetry Breaking in Graphs.'' By restricting what kinds of colorings are considered, many variations of distinguishing numbers have been studied. In this paper, we study proper list-colorings of graphs which are also distinguishing and investigate the choice-distinguishing number $\text{ch}_D(G)$ of a graph $G$. Primarily, we focus on the choice-distinguishing number of Cartesian products of graphs. We determine the exact value of $\text{ch}_D(G)$ for lattice graphs and prism graphs and provide an upper bound on the choice-distinguishing number of the Cartesian products of two relatively prime graphs, assuming a sufficient condition is satisfied. We use this result to bound the choice distinguishing number of toroidal grids and the Cartesian product of a tree with a clique. We conclude with a discussion on how, depending on the graphs $G$ and $H$, we may weaken the sufficient condition needed to bound $\text{ch}_D(G\square H)$. Kaul, Hemanshu Ellis, Robert 2022 Thesis application/pdf islandora:1024962 http://hdl.handle.net/10560/islandora:1024962 Illinois Institute of Technology MATH / Applied Mathematics en In Copyright http://rightsstatements.org/page/InC/1.0/ Restricted Access