Perfect colorings of the infinite circulant graph with distances 1 and 2

A coloring of the vertex set in a graph is called perfect if all its identically colored vertices have identical multisets of colors of their neighbors. Refer as the infinite circulant graph with continuous set of $n$ distances to the Cayley graph of the group $\mathbb{Z}$ with generator set $\{1,2,\ldots,n\}$. We obtain a description of all perfect colorings with an arbitrary number of colors of this graph with distances $1$ and $2$. In 2015, there was made a conjecture characterizing perfect colorings for the infinite circulant graphs with a continuous set of $n$ distances. The obtained result confirms the conjecture for $n = 2$. The problem is still open in the case of $n > 2$. Bibliogr. 12.