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 Sibirsk. Mat. Zh., 2011, Volume 52, Number 1, Pages 115–132 (Mi smj2182)

On periodicity of perfect colorings of the infinite hexagonal and triangular grids

S. A. Puzyninaab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b University of Turku, Finland

Abstract: A coloring of vertices of a graph $G$ is called $r$-perfect, if the color structure of each ball of radius $r$ in $G$ depends only on the color of the center of the ball. The parameters of a perfect coloring are given by the matrix $A=(a_{ij})^n_{i,j=1}$, where $n$ is the number of colors and $a_{ij}$ is the number of vertices of color $j$ in a ball centered at a vertex of color $i$. We study the periodicity of perfect colorings of the graphs of the infinite hexagonal and triangular grids. We prove that for every 1-perfect coloring of the infinite triangular and every 1- and 2-perfect coloring of the infinite hexagonal grid there exists a periodic perfect coloring with the same matrix. The periodicity of perfect colorings of big radii have been studied earlier.

Keywords: perfect coloring, equitable partition, infinite graph, hexagonal grid, triangular grid, periodicity.

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English version:
Siberian Mathematical Journal, 2011, 52:1, 91–104

Bibliographic databases:

Document Type: Article
UDC: 519.17
Revised: 15.11.2010

Citation: S. A. Puzynina, “On periodicity of perfect colorings of the infinite hexagonal and triangular grids”, Sibirsk. Mat. Zh., 52:1 (2011), 115–132; Siberian Math. J., 52:1 (2011), 91–104

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. V. Avgustinovich, D. S. Krotov, A. Yu. Vasil'eva, “Completely regular codes in the infinite hexagonal grid”, Sib. elektron. matem. izv., 13 (2016), 987–1016
2. J. Braun, D. Cruz, N. Jonoska, “Platform color designs for interactive molecular arrangements”, Unconventional Computation and Natural Computation, Ucnc 2017, Lecture Notes in Computer Science, 10240, eds. M. Patitz, M. Stannett, Springler, 2017, 69–81
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