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Perfect colorings of the infinite circulant graph with distances 1 and 2

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Abstract

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 ℤ with generator set {1, 2,..., 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.

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Authors and Affiliations

  1. Marshal Budyonny Military Academy of Telecommunications, Tikhoretskii pr. 3, St. Petersburg, 194064, Russia

    M. A. Lisitsyna

  2. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    O. G. Parshina

  3. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, F-69622, Villeurbanne Cedex, France

    O. G. Parshina

Authors
  1. M. A. Lisitsyna
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  2. O. G. Parshina
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Correspondence to M. A. Lisitsyna.

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Original Russian Text © M.A. Lisitsyna, O.G. Parshina, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 3, pp. 20–34.

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Lisitsyna, M.A., Parshina, O.G. Perfect colorings of the infinite circulant graph with distances 1 and 2. J. Appl. Ind. Math. 11, 381–388 (2017). https://doi.org/10.1134/S1990478917030097

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  • DOI: https://doi.org/10.1134/S1990478917030097

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