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 ) ni,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.
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Original Russian Text Copyright © 2011 Puzynina S. A.
The author was supported in part by the Russian Foundation for Basic Research (Grants 10-01-00424 and 09-01-00244), the Federal Target Program “Scientific and Educational Personnel of Innovation Russia” for 2009–2013 (State Contract 02.740.11.0429), and the Finnish Cultural Foundation.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 1, pp. 115–132, January–February, 2011.
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Puzynina, S.A. On periodicity of perfect colorings of the infinite hexagonal and triangular grids. Sib Math J 52, 91–104 (2011). https://doi.org/10.1134/S0037446606010101
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DOI: https://doi.org/10.1134/S0037446606010101