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Sib. Èlektron. Mat. Izv., 2010, Volume 7, Pages 65–75 (Mi semr228)  

This article is cited in 5 scientific papers (total in 5 papers)

Research papers

On perfect $2$-colorings of the hypercube

K. V. Vorobeva, D. G. Fon-Der-Flaassb

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

Abstract: A vertex coloring of a graph is called perfect if the multiset of colors appearing on the neighbours of any vertex depends only on the color of the vertex. The parameters of a perfect coloring are thus given by a $n\times n$ matrix, where $n$ is the number of colors.
We give a recursive construction which can produce many different perfect colorings of the hypercube $H_n $ with $2$ colors and the parameters $({
\begin{array}{ll} a & b
c & d \end{array}
})$ satisfying the conditions $({b,c})=1,b+c=2^m$, $c>1$. In particular, this construction allows one to find many non-isomorphic perfect colorings with the parameters $( {
\begin{array}{ll} k\cdot a & k\cdot b k\cdot c & k\cdot d \end{array}
})$. For the parameters $({
\begin{array}{ll} a & b
c & d \end{array}
})$ satisfying the extra condition $a\ge c-({b,c})$, we find a lower bound on the number of produced colorings which is hyperexponential in $n$.

Keywords: Hypercube, perfect coloring, perfect code.

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Bibliographic databases:

Document Type: Article
UDC: 517.95
MSC: 76S05
Received December 22, 2009, published March 10, 2010

Citation: K. V. Vorobev, D. G. Fon-Der-Flaass, “On perfect $2$-colorings of the hypercube”, Sib. Èlektron. Mat. Izv., 7 (2010), 65–75

Citation in format AMSBIB
\Bibitem{VorFon10}
\by K.~V.~Vorobev, D.~G.~Fon-Der-Flaass
\paper On perfect $2$-colorings of the hypercube
\jour Sib. \`Elektron. Mat. Izv.
\yr 2010
\vol 7
\pages 65--75
\mathnet{http://mi.mathnet.ru/semr228}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2610166}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. N. Potapov, “O sovershennykh raskraskakh buleva $n$-kuba i korrelyatsionno-immunnykh funktsiyakh maloi plotnosti”, Sib. elektron. matem. izv., 7 (2010), 372–382  mathnet
    2. S. V. Avgustinovich, M. A. Lisitsyna, “Perfect 2-colorings of transitive cubic graphs”, J. Appl. Industr. Math., 5:4 (2011), 519–528  mathnet  crossref  mathscinet  zmath
    3. S. V. Avgustinovich, A. Yu. Vasil'eva, I. V. Sergeeva, “Distance regular colorings of the infinite rectangular grid”, J. Appl. Industr. Math., 6:3 (2012), 280–285  mathnet  crossref  mathscinet  zmath
    4. K. V. Vorobev, “Kratnye sovershennye kody v giperkube”, Diskretn. analiz i issled. oper., 19:4 (2012), 60–65  mathnet  mathscinet
    5. V. N. Potapov, “O bulevykh funktsiyakh, pochti uravnoveshennykh v granyakh”, PDM. Prilozhenie, 2012, no. 5, 23–25  mathnet
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