A new sufficient condition for a 2-strong digraph to be Hamiltonian
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909280157827072 |
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| author | Darbinyan, Samvel Kh. |
| author_facet | Darbinyan, Samvel Kh. |
| contents | In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian:
{\it Let $D$ be a 2-strong digraph of order $n\geq 9$.
If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}.
This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian.
We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_16826 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A new sufficient condition for a 2-strong digraph to be Hamiltonian Darbinyan, Samvel Kh. Combinatorics In this paper we prove the following new sufficient condition for a digraph to be Hamiltonian: {\it Let $D$ be a 2-strong digraph of order $n\geq 9$. If $n-1$ vertices of $D$ have degrees at least $n+k$ and the remaining vertex has degree at least $n-k-4$, where $k$ is a non-negative integer, then $D$ is Hamiltonian}. This is an extension of Ghouila-Houri's theorem for 2-strong digraphs and is a generalization of an early result of the author (DAN Arm. SSR (91(2):6-8, 1990). The obtained result is best possible in the sense that for $k=0$ there is a digraph of order $n=8$ (respectively, $n=9$) with the minimum degree $n-4=4$ (respectively, with the minimum $n-5=4$) whose $n-1$ vertices have degrees at least $n-1$, but it is not Hamiltonian. We also give a new sufficient condition for a 3-strong digraph to be Hamiltonian-connected. |
| title | A new sufficient condition for a 2-strong digraph to be Hamiltonian |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.16826 |