On Hamiltonian bypasses in digraphs and bipartite digraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908458757914624 |
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| author | Darbinyan, Samvel Kh. |
| author_facet | Darbinyan, Samvel Kh. |
| contents | A Hamiltonian path in a digraph $D$ in which the initial vertex dominates the terminal vertex is called a Hamiltonian bypass. Let $D$ be a 2-strong digraph of order $p\geq 3$ and let $z$ be some vertex of $D$. Suppose that every vertex of $D$ other than $z$ has degree at least $p$. We introduce and study a conjecture which claims that there exists a smallest integer $k$ such that if $d(z)\geq k$, then $D$ contains a Hamiltonian bypass. In this paper, we prove: (i) If $D$ is Hamiltonian or $z$ has a degree greater than $(p-1)/3$,
then $D$ contains a Hamiltonian bypass. (ii) If a strong balanced bipartite digraph $B$ of order $2a\geq 6$ satisfies the condition that $d^+(u)+d^-(v)\geq a+1$ for all vertices $u$ and $v$ from different partite sets such that $B$ does not contain the arc $uv$, then $B$ contains a Hamiltonian bypass. Furthermore, the lower bound $a+1$ is sharp. The first result improves a result of Benhocine (J. of Graph Theory, 8, 1984) and a result of the author (Math. Problems of Computer Science,
54, 2020)
We also suggest some conjectures and problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15435 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Hamiltonian bypasses in digraphs and bipartite digraphs Darbinyan, Samvel Kh. Combinatorics A Hamiltonian path in a digraph $D$ in which the initial vertex dominates the terminal vertex is called a Hamiltonian bypass. Let $D$ be a 2-strong digraph of order $p\geq 3$ and let $z$ be some vertex of $D$. Suppose that every vertex of $D$ other than $z$ has degree at least $p$. We introduce and study a conjecture which claims that there exists a smallest integer $k$ such that if $d(z)\geq k$, then $D$ contains a Hamiltonian bypass. In this paper, we prove: (i) If $D$ is Hamiltonian or $z$ has a degree greater than $(p-1)/3$, then $D$ contains a Hamiltonian bypass. (ii) If a strong balanced bipartite digraph $B$ of order $2a\geq 6$ satisfies the condition that $d^+(u)+d^-(v)\geq a+1$ for all vertices $u$ and $v$ from different partite sets such that $B$ does not contain the arc $uv$, then $B$ contains a Hamiltonian bypass. Furthermore, the lower bound $a+1$ is sharp. The first result improves a result of Benhocine (J. of Graph Theory, 8, 1984) and a result of the author (Math. Problems of Computer Science, 54, 2020) We also suggest some conjectures and problems. |
| title | On Hamiltonian bypasses in digraphs and bipartite digraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.15435 |