The perfect colorings of circulant graphs with a large number of colors

An infinite circulant graph with a continuous set of distances is a graph, whose set of vertices is the set of integers, and two vertices $i$ and $j$ are adjacent if $|i-j| \in \{1,2,…,n\}$. We study perfect colorings of such graph with $k$ colors for $k$ at least $3n+3$. A complete description of them is obtained.