On the Locating Chromatic Number of the Cartesian Product of Graphs

Let c be a proper k-coloring of a connected graph G and Π = (V1, V2, . . . , Vk) 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 Π is defined to be the ordered k-tuple cΠ(v) := (d(v, V1), d(v, V2), . . . , d(v, Vk)), where d(v, Vi) = min{d(v, x) | x ∈ Vi}, 1 ≤ i ≤ 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 χ L (G). In this paper, we study the locating chromatic numbers of grids, the cartesian product of paths and complete graphs, and the cartesian product of two complete graphs.