A generalization of the Hamiltonian cycle in dense digraphs

TL;DR

This paper proves that dense digraphs with high minimum in- and out-degree contain the reverse square of a Hamiltonian cycle, extending previous results in the field.

Contribution

It generalizes the existence of the reverse square of Hamiltonian cycles in dense digraphs with high minimum degree.

Findings

Every sufficiently large dense digraph contains the reverse square of a Hamiltonian cycle.

The minimum in- and out-degree threshold is at least (2/3 + γ)n for some γ > 0.

The result extends prior work by Czygrinow, Kierstead, and Molla.

Abstract

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.