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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.10458 |
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Sommario:
- 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.