On Locally Identifying Coloring of Cartesian Product and Tensor Product of Graphs

For a positive integer $k$, a proper $k$-coloring of a graph $G$ is a mapping $f: V(G) \rightarrow \{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for each edge $uv \in E(G)$. The smallest integer $k$ for which there is a proper $k$-coloring of $G$ is called chromatic number of $G$, denoted by $\chi(G)$. A \emph{locally identifying coloring} (for short, lid-coloring) of a graph $G$ is a proper $k$-coloring of $G$ such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer $k$ such that $G$ has a lid-coloring with $k$ colors is called \emph{locally identifying chromatic number} (for short, \emph{lid-chromatic number}) of $G$, denoted by $\chi_{lid}(G)$. In this paper, we study lid-coloring of Cartesian product and tensor product of two graphs. We prove that if $G$ and $H$ are two connected graphs having at least two vertices then (a) $\chi_{lid}(G \square H) \leq \chi(G) \chi(H)-1$ and (b) $\chi_{lid}(G \times H) \leq \chi(G) \chi(H)$. Here $G \square H$ and $G \times H$ denote the Cartesian and tensor products of $G$ and $H$ respectively. We also give exact values of lid-chromatic number of Cartesian product (resp. tensor product) of two paths, a cycle and a path, and two cycles.