Let us consider now a ${\mathsf{N}}$-cube where an Hamiltonian
cycle is removed.
Let $f$ be the corresponding function.
-The question which remains to solve is
-can we always find $b$ such that $\Gamma_{\{b\}}(f)$ is strongly connected.
+The question which remains to solve is:
+\emph{can we always find $b$ such that $\Gamma_{\{b\}}(f)$ is strongly connected?}
-The answer is indeed positive. We furtheremore have the following strongest
+The answer is indeed positive. We furthermore have the following strongest
result.
\begin{thrm}
There exist $b \in \Nats$ such that $\Gamma_{\{b\}}(f)$ is complete.
graph $\Gamma_{\{b\}}(f)$ if and only if $M^b_{xy}$ is positive
where $M$ is the Markov matrix of $\Gamma(f)$.
It has been shown in~\cite[Lemma 3]{bcgr11:ip} that $M$ is regular.
-There exists thus $b$ such there is an arc between any $x$ and $y$.
+Thus, there exists $b$ such that there is an arc between any $x$ and $y$.
\end{proof}
This section ends with the idea of removing a Hamiltonian cycle in the
In such a context, the Hamiltonian cycle is equivalent to a Gray code.
Many approaches have been proposed a way to build such codes, for instance
the Reflected Binary Code. In this one, one of the bits is switched
-exactly $2^{\mathsf{N}-}$ for a $\mathsf{N}$-length cycle.
+exactly $2^{\mathsf{N}-}$ \ANNOT{formule incomplète : $2^{\mathsf{N}-1}$ ??} for a $\mathsf{N}$-length cycle.
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The function that is built
-from the
+from the \ANNOT{Phrase non terminée}
The next section presents how to build balanced Hamiltonian cycles in the
$\mathsf{N}$-cube with the objective to embed them into the
