Some results on minimum saturated graphs
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914089481011200 |
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| author | Zhang, Chenke Cui, Qing Hu, Jinze Yue, Erfei Ji, Shengjin |
| author_facet | Zhang, Chenke Cui, Qing Hu, Jinze Yue, Erfei Ji, Shengjin |
| contents | Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\overline{G})$, $G+e$ creates a copy of some member in $ \mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges of an $\mathcal{F}$-saturated graphs with $n$ vertices, denoted by $\sat(n,\mathcal{F})$. If $\mathcal{F}=\{F\}$, then we write it as $\sat(n,F)$ for short. In this paper, we determine the exact value of $\sat(n,\{K_3,P_k\})$, and as its application, we obtain two bounds of $\sat(n,K_3\cup P_k)$ for $k\ge 10$ and sufficiently large $n$. Furthermore, $\sat(n,K_1\lor F)$ is determined, where $F$ is a linear forest without isolated vertices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_10458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some results on minimum saturated graphs Zhang, Chenke Cui, Qing Hu, Jinze Yue, Erfei Ji, Shengjin Combinatorics Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\overline{G})$, $G+e$ creates a copy of some member in $ \mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges of an $\mathcal{F}$-saturated graphs with $n$ vertices, denoted by $\sat(n,\mathcal{F})$. If $\mathcal{F}=\{F\}$, then we write it as $\sat(n,F)$ for short. In this paper, we determine the exact value of $\sat(n,\{K_3,P_k\})$, and as its application, we obtain two bounds of $\sat(n,K_3\cup P_k)$ for $k\ge 10$ and sufficiently large $n$. Furthermore, $\sat(n,K_1\lor F)$ is determined, where $F$ is a linear forest without isolated vertices. |
| title | Some results on minimum saturated graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.10458 |