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 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)∪ : the vertices\ x, y∈ V(C)\ and the distance fromx$ to $y$ on $C$ is $2\. In this paper, we show that for any real number γ>0 there exists a constant n₀=n₀(γ), such that every digraph on n≥ n₀ vertices with the minimum in- and out-degree at least (2/3+γ)n contains the reverse square of a Hamiltonian cycle. Our result extends a result of Czygrinow, Kierstead and Molla.