This article is cited in
8 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
$\left({
\begin{array}{ll}
a & b\\c & d
\end{array}
}\right)$ 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
$\left(
{
\begin{array}{ll}
k\cdot a & k\cdot b \\ k\cdot c & k\cdot d
\end{array}
}\right)$.
For the parameters $\left({
\begin{array}{ll}
a & b\\c & d
\end{array}
}\right)$ 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.
UDC:
517.95
MSC: 76S05 Received December 22, 2009, published
March 10, 2010