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 $f:V\left(G\right)\to \{1,2,\dots ,k\}$ such that $f\left(u\right)\ne f\left(v\right)$ for each edge $uv$ of $G$. The smallest integer $k$ for which there is a proper $k$-coloring of $G$ is called the chromatic number of $G$, denoted by $\chi \left(G\right)$. A 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 locally identifying chromatic number (for short, lid-chromatic number) of $G$, denoted by ${\chi }_{lid}\left(G\right)$.

This paper studies the lid-coloring of the 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) ${\chi }_{lid}(G\square H)\le \chi \left(G\right)\chi \left(H\right)-1$ and (b) ${\chi }_{lid}(G\times H)\le \chi \left(G\right)\chi \left(H\right)$. Here $G\square H$ and $G\times H$ denote the Cartesian and tensor products of $G$ and $H$, respectively. We determine the lid-chromatic numbers of $C_m\square P_n$, $C_m\square C_n$, $P_m\times P_n$, $C_m\times P_n$ and $C_m\times C_n$, where $C_m$ and $P_n$ denote a cycle and a path on $m$ and $n$ vertices, respectively.