Given a graph $G=\big{(}V(G),E(G)\big{)}$, a mapping from $V(G)$ to $\{1,\cdots,|V(G)|\}$ (the numbers $1,\cdots,|V(G)|$ are usually called colors) is said to be a $2$-distance coloring of $G$ if any two vertices at distance at most two from each other receive different colors. Such a mapping with the minimum number of colors is said to be an optimal $2$-distance coloring of $G$. The $2$-distance chromatic number $\chi_{2}(G)$ of a graph $G$ is the number of colors assigned by an optimal $2$-distance coloring to the vertices of $G$. $\lozenge$ In this paper, we continue the study of this classic topic in graph theory. The main focus of this work is given to this parameter in some graph products, where we investigate this type of coloring in the strong, direct and rooted products. In particular, in the case of rooted products (in its most general case) we give an exact formula for the $2$-distance chromatic number. This improves the results in a previous paper bounding this parameter from below and above in a special case. We next give bounds on this parameter for general graphs as well as the exact values for it in some well-known families of graphs.
