On the Locating Chromatic Number of the Cartesian Product of Graphs

Let $c$ be a proper $k$-coloring of a connected graph $G$ and $\mathrm{\Pi }=(C_1,C_2,\mathrm{.}\mathrm{.}\mathrm{.},C_k)$ be an ordered partition of $V(G)$ into the resulting color classes. For a vertex $v$ of $G$, the color code of $v$ with respect to $\mathrm{\Pi }$ is defined to be the ordered $k$-tuple $c_{{}_{\mathrm{\Pi }}}(v):=(d(v,C_1),d(v,C_2),\mathrm{.}\mathrm{.}\mathrm{.},d(v,C_k)),$ where $d(v,C_i)=\min{d(v,x) | x\in C_i}, 1\leq i\leq k$. If distinct vertices have distinct color codes, then $c$ is called a locating coloring. The minimum number of colors needed in a locating coloring of $G$ is the locating chromatic number of $G$, denoted by ${\text{Cchi}}_{{}_L}(G)$. In this paper, we study the locating chromatic number of grids, the cartesian product of paths and complete graphs, and the cartesian product of two complete graphs.