Geodesic
Perfect colorings of the infinite circulant graph with distances~1 and~2
Diskretnyj analiz i issledovanie operacij, Tome 24 (2017) no. 3, pp. 20-34

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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.
Keywords: perfect coloring
Mots-clés : circulant graph, equitable partition. 
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     title = {Perfect colorings of the infinite circulant graph with distances~1 and~2},
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M. A. Lisitsyna; O. G. Parshina. Perfect colorings of the infinite circulant graph with distances~1 and~2. Diskretnyj analiz i issledovanie operacij, Tome 24 (2017) no. 3, pp. 20-34. http://geodesic.mathdoc.fr/item/DA_2017_24_3_a1/