  A vertex coloring of a given simple graph $G=(V,E)$ with $k$ colors
($k$-coloring) is a map from its vertex set to the set of integers
$\{1,2,3,\dots, k\}$. A coloring is called perfect if the multiset of colors
appearing on the neighbours of any vertex depends only on the color of the
vertex. We consider perfect colorings of Cayley graphs of the additive group of
integers with generating set $\{1,-1,3,-3,5,-5,\dots, 2n-1,1-2n\}$ for a
positive integer $n$. We enumerate perfect $2$-colorings of the graphs under
consideration and state the conjecture generalizing the main result to an
arbitrary number of colors.
