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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2506.08810 |
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| _version_ | 1866916927016796160 |
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| author | Bonamy, Marthe Groenland, Carla Johnston, Tom Morrison, Natasha Scott, Alex |
| author_facet | Bonamy, Marthe Groenland, Carla Johnston, Tom Morrison, Natasha Scott, Alex |
| contents | A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_08810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinite induced-saturated graphs Bonamy, Marthe Groenland, Carla Johnston, Tom Morrison, Natasha Scott, Alex Combinatorics A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$. |
| title | Infinite induced-saturated graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.08810 |