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 $C_m \square P_n$, $C_m \square C_n$, $P_m \times P_n$, $C_m \times P_n$ and $C_m \times C_n$, where $C_m$ and $P_n$ denote a cycle and a path on $m$ and $n$ vertices respectively.

翻译：对于正整数$k$，图$G$的一个正常$k$-染色是映射$f: V(G) \rightarrow \{1,2, \ldots, k\}$，使得对$G$的每条边$uv$均有$f(u) \neq f(v)$。使$G$存在正常$k$-染色的最小整数$k$称为图$G$的色数，记为$\chi(G)$。图$G$的局部识别染色（简称lid-染色）是一种正常$k$-染色，使得任意一对具有不同闭邻域的相邻顶点，在其闭邻域内的颜色集合互异。使$G$存在$k$种颜色的lid-染色的最小整数$k$称为图$G$的局部识别色数（简称lid-色数），记为$\chi_{lid}(G)$。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$H$是至少有两个顶点的连通图，则(a) $\chi_{lid}(G \square H) \leq \chi(G) \chi(H)-1$ 且(b) $\chi_{lid}(G \times H) \leq \chi(G) \chi(H)$。这里$G \square H$和$G \times H$分别表示$G$与$H$的笛卡尔积和张量积。我们确定了$C_m \square P_n$、$C_m \square C_n$、$P_m \times P_n$、$C_m \times P_n$和$C_m \times C_n$的lid-色数，其中$C_m$和$P_n$分别表示包含$m$个和$n$个顶点的圈与路。