A generalization of the Hamiltonian cycle in dense digraphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909269604958208 |
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| author | Zhang, Jie Wang, Zhilan Yan, Jin |
| author_facet | Zhang, Jie Wang, Zhilan Yan, Jin |
| contents | 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 $γ>0$ there exists a constant $n_0=n_0(γ)$, such that every digraph on $n\geq n_0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18636 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of the Hamiltonian cycle in dense digraphs Zhang, Jie Wang, Zhilan Yan, Jin Combinatorics 05C20, 05C70, 05C07 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 $γ>0$ there exists a constant $n_0=n_0(γ)$, such that every digraph on $n\geq n_0$ 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. |
| title | A generalization of the Hamiltonian cycle in dense digraphs |
| topic | Combinatorics 05C20, 05C70, 05C07 |
| url | https://arxiv.org/abs/2407.18636 |