A generalization of the Hamiltonian cycle in dense digraphs

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 defined as the minimum length of a directed path from $x$ to $y$. The square of the cycle $C$ is the graph with vertex set $V\left(C\right)$, where 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 vertex set $V\left(C\right)$, and arc set $A\left(C\right)\cup \{yx:\text{the vertices}x,y\in V\left(C\right)\text{and the distance from}x\text{to}y\text{in}C\text{is}2\}$. In this paper, we prove that for every real number $\gamma >0$, there exists a positive integer $n_0=n_0\left(\gamma \right)$ such that every digraph on $n\ge n_0$ vertices with the minimum in- and out-degree at least $(2/3+\gamma )n$ contains the reverse square of a Hamiltonian cycle. This result generalizes a theorem of Czygrinow, Kierstead and Molla.