A lower bound on the saturation number and a strengthening for triangle-free graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913961296789504 |
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| author | Buchanan, Calum Rombach, Puck |
| author_facet | Buchanan, Calum Rombach, Puck |
| contents | The saturation number $\operatorname{sat}(n, H)$ of a graph $H$ and positive integer $n$ is the minimum size of a graph of order $n$ which does not contain a subgraph isomorphic to $H$ but to which the addition of any edge creates such a subgraph. Erdős, Hajnal, and Moon first studied saturation numbers of complete graphs, and Cameron and Puleo introduced a general lower bound on $\operatorname{sat}(n,H)$. In this paper, we present another lower bound on $\operatorname{sat}(n, H)$ with strengthenings for graphs $H$ in several classes, all of which include the class of triangle-free graphs. Demonstrating its effectiveness, we determine the saturation numbers of diameter-$3$ trees up to an additive constant; these are double stars $S_{s,t}$ of order $s + t$ whose central vertices have degrees $s$ and $t$. Faudree, Faudree, Gould, and Jacobson determined that $\operatorname{sat}(n, S_{t,t}) = (t-1)n/2 + O(1)$. We prove that $\operatorname{sat}(n,S_{s,t}) = (st+s)n/(2t+4) + O(1)$ when $s < t$. We also apply our lower bound to caterpillars and demonstrate an upper bound on the saturation numbers of certain diameter-$4$ caterpillars. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lower bound on the saturation number and a strengthening for triangle-free graphs Buchanan, Calum Rombach, Puck Combinatorics 05C35 The saturation number $\operatorname{sat}(n, H)$ of a graph $H$ and positive integer $n$ is the minimum size of a graph of order $n$ which does not contain a subgraph isomorphic to $H$ but to which the addition of any edge creates such a subgraph. Erdős, Hajnal, and Moon first studied saturation numbers of complete graphs, and Cameron and Puleo introduced a general lower bound on $\operatorname{sat}(n,H)$. In this paper, we present another lower bound on $\operatorname{sat}(n, H)$ with strengthenings for graphs $H$ in several classes, all of which include the class of triangle-free graphs. Demonstrating its effectiveness, we determine the saturation numbers of diameter-$3$ trees up to an additive constant; these are double stars $S_{s,t}$ of order $s + t$ whose central vertices have degrees $s$ and $t$. Faudree, Faudree, Gould, and Jacobson determined that $\operatorname{sat}(n, S_{t,t}) = (t-1)n/2 + O(1)$. We prove that $\operatorname{sat}(n,S_{s,t}) = (st+s)n/(2t+4) + O(1)$ when $s < t$. We also apply our lower bound to caterpillars and demonstrate an upper bound on the saturation numbers of certain diameter-$4$ caterpillars. |
| title | A lower bound on the saturation number and a strengthening for triangle-free graphs |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2402.11387 |