The saturation number of $K^s_t$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914542995374080 |
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| author | Zhao, Xinghui You, Lihua Zhang, Xiaoxue |
| author_facet | Zhao, Xinghui You, Lihua Zhang, Xiaoxue |
| contents | For a given graph $F$, a graph $G$ is said to be $F$-saturated if $G$ contains no copy of $F$ but for any edge $uv\notin E(G)$, $G+uv$ contains a copy of $F$. The saturation number $sat(n,F)$ is defined as the minimum number of edges among all $n$-vertex $F$-saturated graphs. The virus graph $K^s_t$, where $s\geq0$ and $t\geq \max\{3,s\}$, is a graph of order $s+t$ constructed by attaching $s$ distinct leaves to $s$ different vertices of a complete graph $K_t$. Hua and Peng [Discrete Math. 349 (2026) 114674] determined $sat(n,K^2_3)$ and characterized its corresponding extremal graphs. In this paper, we determine $sat(n,K^3_3)$ and $sat(n,K^2_t)$ with $t\geq 4$, together with the structural descriptions of the related extremal saturated graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_07179 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The saturation number of $K^s_t$ Zhao, Xinghui You, Lihua Zhang, Xiaoxue Combinatorics 05C35, 05C70 For a given graph $F$, a graph $G$ is said to be $F$-saturated if $G$ contains no copy of $F$ but for any edge $uv\notin E(G)$, $G+uv$ contains a copy of $F$. The saturation number $sat(n,F)$ is defined as the minimum number of edges among all $n$-vertex $F$-saturated graphs. The virus graph $K^s_t$, where $s\geq0$ and $t\geq \max\{3,s\}$, is a graph of order $s+t$ constructed by attaching $s$ distinct leaves to $s$ different vertices of a complete graph $K_t$. Hua and Peng [Discrete Math. 349 (2026) 114674] determined $sat(n,K^2_3)$ and characterized its corresponding extremal graphs. In this paper, we determine $sat(n,K^3_3)$ and $sat(n,K^2_t)$ with $t\geq 4$, together with the structural descriptions of the related extremal saturated graphs. |
| title | The saturation number of $K^s_t$ |
| topic | Combinatorics 05C35, 05C70 |
| url | https://arxiv.org/abs/2605.07179 |