  Let D be a digraph and C be a cycle in D. For any two vertices x and y in D,
the distance from x to y is the minimum length of a path from x to y. We denote
the square of Let $D$ be a digraph and $C$ be a cycle in $D$. For any two
vertices $x$ and $y$ in $D$, the distance from $x$ to $y$ is the minimum length
of a path from $x$ to $y$. We denote the square of the cycle $C$ to be the
graph whose vertex set is $V(C)$ and for distinct vertices $x$ and $y$ in $C$,
there is an arc from $x$ to $y$ if and only if the distance from $x$ to $y$ in
$C$ is at most $2$. The reverse square of the cycle $C$ is the digraph with the
same vertex set as $C$, and the arc set $A(C)\cup \{yx: \mbox{the vertices}\ x,
y\in V(C)\ \mbox{and the distance from $x$ to $y$ on $C$ is $2$}\}$. In this
paper, we show that for any real number $\gamma>0$ there exists a constant
$n_0=n_0(\gamma)$, such that every digraph on $n\geq n_0$ vertices with the
minimum in- and out-degree at least $(2/3+\gamma)n$ contains the reverse square
of a Hamiltonian cycle. Our result extends a result of Czygrinow, Kierstead and
Molla.
