English version:
Journal of Applied and Industrial Mathematics, 2017, 11:3, 381-388

Volume 24, No 3, 2017, P. 20-34

UDC 519.174.7
M. A. Lisitsyna and O. G. Parshina
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 {$1, 2, \dots , n$}. 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, circulant graph, equitable partition.

DOI: 10.17377/daio.2017.24.559

Mariya A. Lisitsyna 1
Olga G. Parshina 2,3

1. Marshal Budyonny Military Academy of Telecommunications,
3 Tikhoretsky Ave., 194064 St. Petersburg, Russia
2. Sobolev Institute of Mathematics,
4 Acad. Koptyug Ave., 630090 Novosibirsk, Russia
3. Institut Camille Jordan, Université Claude Bernard Lyon 1,
43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France
e-mail: lisicinama@ngs.ru, parolja@gmail.com

Received 2 December 2016
Revised 30 March 2017


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 © Sobolev Institute of Mathematics, 2015