Geodesic
Completely regular codes in the infinite hexagonal grid
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 987-1016.

Voir la notice de l'article provenant de la source Math-Net.Ru

A set $C$ of vertices of a simple graph is called a completely regular code if for each $i=0$, $1$, $2, \ldots$ and $j = i-1$, $i$, $i+1$, all vertices at distance $i$ from $C$ have the same number $s_{ij}$ of neighbors at distance $j$ from $C$. We characterize the completely regular codes in the infinite hexagonal grid graph.
Keywords: completely regular code, perfect coloring
Mots-clés : equitable partition, partition design, hexagonal grid. 
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S. V. Avgustinovich; D. S. Krotov; A. Yu. Vasil'eva. Completely regular codes in the infinite hexagonal grid. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 13 (2016), pp. 987-1016. http://geodesic.mathdoc.fr/item/SEMR_2016_13_a64/

[1] S. V. Avgustinovich, A. Yu. Vasil'eva, “Distance regular colorings of $n$-dimensional rectangular grid”, Proc. Thirteenth Int. Workshop on Algebraic and Combinatorial Coding Theory, ACCT 2012 (Pomorie, Bulgaria, June 2012), 93–98 http://www.moi.math.bas.bg/moiuser/\allowbreakÃCCT2012/b6.pdf

[2] Diskretn. Anal. Issled. Oper., 18:2 (2011), 3–10 | DOI | MR | MR | Zbl

[3] M. A. Axenovich, “On multiple coverings of the infinite rectangular grid with balls of constant radius”, Discrete Math., 268:1–3 (2003), 31–48 | DOI | MR | Zbl

[4] A. E. Brouwer, A. M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989 | DOI | MR | Zbl

[5] B. Grüenbaum, G. C. Shephard, Tilings and Patterns, 1 edition, W. H. Freeman and Company, 1987 | MR | Zbl

[6] D. S. Krotov, Perfect colorings of $Z^2$: Nine colors, 2009., arXiv: 0901.0004 [math.CO]

[7] D. S. Krotov, “On weight distributions of perfect colorings and completely regular codes”, Des. Codes Cryptography, 61:3 (2011), 315–329 | DOI | MR | Zbl

[8] D. P. Plotnikov, Perfect colorings of the vertices of a regular graph of degree three into three and more colors, Graduate thesis, Novosibirsk State University, 2005 (In Russian)

[9] S. A. Puzynina, “Periodicity of perfect colorings of the infinite rectangular grid”, Diskretn. Anal. Issled. Oper., Ser. 1, 11:1 (2004), 79–92 (In Russian) | MR | Zbl

[10] S. A. Puzynina, “The perfect colorings of the vertices of the graph $G(Z^2)$ into three colors”, Diskretn. Anal. Issled. Oper., Ser. 2, 12:1 (2005), 37–54 (In Russian) | MR | Zbl

[11] Sib. Mat. Zh., 52:1 (2011), 115–132 | DOI | MR | Zbl

[12] A. Yu. Vasil'eva, “Distance regular colorings of the infinite triangular grid”, International Conference “Mal'tsev Meeting”, Collection of Abstracts (November 10–13, 2014), Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk, 2014, 98 http://www.math.nsc.ru/conference/malmeet/14/Malmeet2014.pdf