Geodesic
The perfect $2$-colorings of infinite circulant graphs with a continuous set of odd distances
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 590-603

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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.
Keywords: perfect coloring, Cayley graph
Mots-clés : circulant graph, equitable partition. 
@article{SEMR_2020_17_a68,
     author = {O. G. Parshina and M. A. Lisitsyna},
     title = {The perfect $2$-colorings of infinite circulant graphs with a continuous set of odd distances},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {590--603},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a68/}
}
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O. G. Parshina; M. A. Lisitsyna. The perfect $2$-colorings of infinite circulant graphs with a continuous set of odd distances. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 590-603. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a68/