The saturation number of wheels
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915252943192064 |
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| author | Qiu, Yanzhe He, Zhen Lu, Mei Xu, Yiduo |
| author_facet | Qiu, Yanzhe He, Zhen Lu, Mei Xu, Yiduo |
| contents | A graph $G$ is said to be $F$-free, if $G$ does not contain any copy of $F$. $G$ is said to be $F$-semi-saturated, if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. $G$ is said to be $F$-saturated, if $G$ is $F$-free and $F$-semi-saturated. The saturation number $sat(n,F)$ (resp. semi-saturation number $ssat(n,F)$) is the minimum number of edges in an $F$-saturated (resp. $F$-semi-saturated) graph of order $n$. In this paper we proved several results on the (semi)-saturation number of the wheel graph $W_k=K_1 \vee C_k$. Let $k,n$ be positive integers with $k \geq 8$ and $n \geq 56k^3$, we showed that $(s)sat(n,W_k)=n-1+(s)sat(n-1,C_k)$. We also establish the lower bound of semi-saturation number of $W_k$ with restriction on maximum degree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_10268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The saturation number of wheels Qiu, Yanzhe He, Zhen Lu, Mei Xu, Yiduo Combinatorics 05C35 A graph $G$ is said to be $F$-free, if $G$ does not contain any copy of $F$. $G$ is said to be $F$-semi-saturated, if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. $G$ is said to be $F$-saturated, if $G$ is $F$-free and $F$-semi-saturated. The saturation number $sat(n,F)$ (resp. semi-saturation number $ssat(n,F)$) is the minimum number of edges in an $F$-saturated (resp. $F$-semi-saturated) graph of order $n$. In this paper we proved several results on the (semi)-saturation number of the wheel graph $W_k=K_1 \vee C_k$. Let $k,n$ be positive integers with $k \geq 8$ and $n \geq 56k^3$, we showed that $(s)sat(n,W_k)=n-1+(s)sat(n-1,C_k)$. We also establish the lower bound of semi-saturation number of $W_k$ with restriction on maximum degree. |
| title | The saturation number of wheels |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2503.10268 |