Consider the set $A$ of the $2^{n}$. binary sequences of length $n$. This exercise concerns the existence of a circular arrangement $\gamma_{n}$ of $2^{n} 0$ s and $1 s$, so that the $2^{n}$ sequences of $n$ consecutive bits of $\gamma$ give all of $A$; that is, are all distinct. Such a circular arrangement is called a de Bruijn cycle. For example, if $n=2$, the circular arrangement $0,0,1,1$. (regarding the first 0 as following the last 1 ) gives 0,$0 ; 0,1 ; 1,1 ;$ and $1,0 .$ For $n=3,0,0,0,1,0,1,1,1$ (regarded cyclically) is a de Bruijn cycle. Define a digraph $\Gamma_{n}$ whose vertices are the $2^{n-1}$ binary sequences of length $n-1$. Given two such binary sequences $x$ and $y$, we put an arc $e$ from $x$ to $y$, provided that the last $n-2$ bits of $x$ agree with the first $n-2$ bits of $y_{1}$ and then we label the arc $e$ with the first. bit of $x$.
(a) Prove that every vertex of $\Gamma_{n}$ has indegree and outdegree equal to 2 . Thus, $\Gamma_{n}$ has a total of $2 \cdot 2^{n-1}=2^{n}$ arcs.
(b) Prove that $\Gamma_{n}$ is strongly connected, and hence $\Gamma_{n}$ has a closed Eulerian directed trail (of length $\left.2^{n}\right)$.
(c) Let $b_{1}, b_{2}, \ldots, b_{2^{n}}$ be the labels of the arcs (considered as a circular arrangement) as we traverse an Eulerian directed trail of $\Gamma_{n}$. Prove that $b_{1}, b_{2}, \ldots, b_{2^{n}}$ is a de Bruijn cycle.
(d) Prove that, given any two vertices $x$ and $y$ of the digraph $\Gamma_{n}$, there is a path from $x$ to $y$ of length at most $n-1$.