4K_1 free graphs on 13 vertices have cop number at most 2
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866908764738682880 |
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| author | Wu, Zhaoyu |
| author_facet | Wu, Zhaoyu |
| contents | The game of cops and robber has been studied for many years. Denoting $\mathsf{Forb}(4K_1)$ to be the family of all graphs that contain no induced subgraph isomorphic to $4K_1$ (e.g., with independence number less than $4$), we prove that for any $G\in\mathsf{Forb}(4K_1)$, we have $c(G)\leq 2$, where $c(\cdot)$ is the cop number. This improves a lower bound of a question proposed by Char et al. in a recent paper (arxiv, 2025), that any counterexample of a conjecture raised by Turcotte (2022) when $p=4$ must have at least 14 vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00917 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 4K_1 free graphs on 13 vertices have cop number at most 2 Wu, Zhaoyu Combinatorics The game of cops and robber has been studied for many years. Denoting $\mathsf{Forb}(4K_1)$ to be the family of all graphs that contain no induced subgraph isomorphic to $4K_1$ (e.g., with independence number less than $4$), we prove that for any $G\in\mathsf{Forb}(4K_1)$, we have $c(G)\leq 2$, where $c(\cdot)$ is the cop number. This improves a lower bound of a question proposed by Char et al. in a recent paper (arxiv, 2025), that any counterexample of a conjecture raised by Turcotte (2022) when $p=4$ must have at least 14 vertices. |
| title | 4K_1 free graphs on 13 vertices have cop number at most 2 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.00917 |