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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.13434 |
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| _version_ | 1866913033291300864 |
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| author | Bae, Ji Ho |
| author_facet | Bae, Ji Ho |
| contents | We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~$10$ and~$11$ vertices under local complementation. At the extremal order $n=10$, exactly six non-isomorphic graphs avoid~$E_4$ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~$8{,}712$. Thus $k=4$ is the first case in which the general upper bound $2^k-1$ is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~$\Rvm(k)$ that surpass the leading term of the asymptotic bound for all $k\leq 9$; in particular, $\Rvm(5)\geq 13$. We also describe structural properties of the six extremal graphs and formulate the next open problem, whether $\Rvm(5)=15$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13434 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex-minor Ramsey numbers: exact values and extremal structure Bae, Ji Ho Combinatorics 05C76, 05C55, 05C85 We determine the vertex-minor Ramsey number $\Rvm(4)=11$, where $\Rvm(k)$ is the smallest~$n$ such that every $n$-vertex graph contains the edgeless graph~$E_k$ as a vertex-minor. We prove this by an exhaustive classification of the graphs on~$10$ and~$11$ vertices under local complementation. At the extremal order $n=10$, exactly six non-isomorphic graphs avoid~$E_4$ as a vertex-minor; up to isomorphism, they represent five LC-equivalence classes, and each labeled LC orbit has cardinality~$8{,}712$. Thus $k=4$ is the first case in which the general upper bound $2^k-1$ is not attained. Using the extremal graphs as building blocks, we derive explicit lower bounds on~$\Rvm(k)$ that surpass the leading term of the asymptotic bound for all $k\leq 9$; in particular, $\Rvm(5)\geq 13$. We also describe structural properties of the six extremal graphs and formulate the next open problem, whether $\Rvm(5)=15$. |
| title | Vertex-minor Ramsey numbers: exact values and extremal structure |
| topic | Combinatorics 05C76, 05C55, 05C85 |
| url | https://arxiv.org/abs/2604.13434 |