Perfect colorings of infinite multifold chains

An infinite chain $C _ {\ inf}$ is a graph whose vertex set coincides with the set of integers, and vertices at a distance of $1$ are connected by edges.
Let $G$ be an arbitrary transitive graph. We paste a copy of the graph $G$ instead of each vertex of the infinite chain, add edges connecting any two vertices from adjacent copies. The resulting graph is called the $G$-fold infinite chain. The graph defined in this way is precisely the lexicographic product of the graphs $C_{\inf}$ and $G$. A complete description of the perfect colorings into arbitrary finite number of colors. is obtained in an of infinite chains multiples of the empty graph on $ n $ vertices. A similar result is obtained for the $K_n$-fold infinite chain.
(Joint work with S. V. Avgustinovich and O. G. Parshina)