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 $uv$ 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 $\chi(G)$. 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 $\chi{lid}(G)$. 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) $\chi{lid}(G \square H) \leq \chi(G) \chi(H)-1$ 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 determine the lid-chromatic number of $Cm \square Pn$, $Cm \square Cn$, $Pm \times Pn$, $Cm \times Pn$ and $Cm \times Cn$, where $Cm$ and $P_n$ denote a cycle and a path on $m$ and $n$ vertices respectively.