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.

翻译：设$k$为正整数，图$G$的一个正常$k$-染色是指映射$f: V(G) \rightarrow \{1,2, \ldots, k\}$，使得对每条边$uv \in E(G)$均有$f(u) \neq f(v)$。使$G$存在正常$k$-染色的最小整数$k$称为$G$的色数，记作$\chi(G)$。图$G$的**局部识别染色**（简称lid-染色）是一种正常$k$-染色，满足任意一对闭邻域不同的相邻顶点，其闭邻域内的颜色集合互异。使$G$存在$k$色lid-染色的最小整数$k$称为$G$的**局部识别色数**（简称lid-色数），记作$\chi_{lid}(G)$。本文研究两个图的笛卡尔积与张量积的lid-染色。我们证明：若$G$和$H$是至少包含两个顶点的连通图，则(a) $\chi_{lid}(G \square H) \leq \chi(G) \chi(H)-1$ 且(b) $\chi_{lid}(G \times H) \leq \chi(G) \chi(H)$。此处$G \square H$与$G \times H$分别表示$G$和$H$的笛卡尔积与张量积。我们还给出了两条路径、一条环与一条路径、以及两条环的笛卡尔积（相应地，张量积）的lid-色数的精确值。