Minimum saturated graphs without $4$-cycles and $5$-cycles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913749683666944 |
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| author | Ma, Yue |
| author_facet | Ma, Yue |
| contents | Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$, but the addition of any edge $e\notin E(G)$ creates at least one copy of some $F\in\mathcal{F}$ within $G$. The minimum size of an $\mathcal{F}$-saturated graph on $n$ vertices is called the saturation number, denoted by $\mbox{sat}(n, \mathcal{F})$. Let $C_r$ be the cycle of length $r$. In this paper, we study on $\mbox{sat}(n, \mathcal{F})$ when $\mathcal{F}$ is a family of cycles. In particular, we determine that $\mbox{sat}(n, \{C_4,C_5\})=\lceil\frac{5n}{4}-\frac{3}{2}\rceil$ for any positive integer $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimum saturated graphs without $4$-cycles and $5$-cycles Ma, Yue Combinatorics Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$, but the addition of any edge $e\notin E(G)$ creates at least one copy of some $F\in\mathcal{F}$ within $G$. The minimum size of an $\mathcal{F}$-saturated graph on $n$ vertices is called the saturation number, denoted by $\mbox{sat}(n, \mathcal{F})$. Let $C_r$ be the cycle of length $r$. In this paper, we study on $\mbox{sat}(n, \mathcal{F})$ when $\mathcal{F}$ is a family of cycles. In particular, we determine that $\mbox{sat}(n, \{C_4,C_5\})=\lceil\frac{5n}{4}-\frac{3}{2}\rceil$ for any positive integer $n$. |
| title | Minimum saturated graphs without $4$-cycles and $5$-cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.16839 |