A neighborhood union condition for the existence of a spanning tree without degree $2$ vertices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915055904227328 |
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| author | Li, Yibo Dong, Fengming Hu, Xiaolan Liu, Huiqing |
| author_facet | Li, Yibo Dong, Fengming Hu, Xiaolan Liu, Huiqing |
| contents | For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree $2$. In this paper, we show that if $G$ is a graph of order $n\ge 270$ and $|N(u)\cup N(v)|\geq\frac{n-1}{2}$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$, then $G$ has a HIST, unless $G$ belongs to three exceptional families of graphs or $G$ has a cut-vertex of degree $2$. This result improves the latest conclusion, due to Ito and Tsuchiya, that a HIST in $G$ can be guaranteed if $d(u)+d(v)\geq n-1$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07128 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A neighborhood union condition for the existence of a spanning tree without degree $2$ vertices Li, Yibo Dong, Fengming Hu, Xiaolan Liu, Huiqing Combinatorics 05C05, 05C10 For a connected graph $G$, a spanning tree $T$ of $G$ is called a homeomorphically irreducible spanning tree (HIST) if $T$ has no vertices of degree $2$. In this paper, we show that if $G$ is a graph of order $n\ge 270$ and $|N(u)\cup N(v)|\geq\frac{n-1}{2}$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$, then $G$ has a HIST, unless $G$ belongs to three exceptional families of graphs or $G$ has a cut-vertex of degree $2$. This result improves the latest conclusion, due to Ito and Tsuchiya, that a HIST in $G$ can be guaranteed if $d(u)+d(v)\geq n-1$ holds for every pair of nonadjacent vertices $u$ and $v$ in $G$. |
| title | A neighborhood union condition for the existence of a spanning tree without degree $2$ vertices |
| topic | Combinatorics 05C05, 05C10 |
| url | https://arxiv.org/abs/2412.07128 |