A generalization of the Hamiltonian cycle in dense digraphs
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.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.18636
- arXiv:
- arXiv:2407.18636
- Bibcode:
- 2024arXiv240718636Z
- Keywords:
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- Mathematics - Combinatorics;
- 05C20;
- 05C70;
- 05C07
- E-Print:
- 14 pages