All Ramsey critical graphs for a large tree versus $tK_{m}$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912444511682560 |
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| author | Cheng, Zhiyu Luo, Zhidan Chen, Pingge |
| author_facet | Cheng, Zhiyu Luo, Zhidan Chen, Pingge |
| contents | Let $H, H_{1}$ and $H_{2}$ be graphs, and let $H\rightarrow (H_{1}, H_{2})$ denote that any red-blue coloring of $E(H)$ yields a red copy of $H_{1}$ or a blue copy of $H_{2}$. The Ramsey number for $H_{1}$ versus $H_{2}$, $r(H_{1}, H_{2})$, is the minimum integer $N$ such that $K_{N}\rightarrow (H_{1}, H_{2})$. The Ramsey critical graph $H$ for $H_{1}$ versus $H_{2}$ is a red-blue edge-colored $K_{N- 1}$ such that $H\not\rightarrow (H_{1}, H_{2})$, where $N= r(H_{1}, H_{2})$. In this paper, we characterize all Ramsey critical graphs for a large tree versus $tK_{m}$. As a corollary, we determine the star-critical Ramsey number for a large tree versus $tK_{m}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18235 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | All Ramsey critical graphs for a large tree versus $tK_{m}$ Cheng, Zhiyu Luo, Zhidan Chen, Pingge Combinatorics 05C55, 05D10 Let $H, H_{1}$ and $H_{2}$ be graphs, and let $H\rightarrow (H_{1}, H_{2})$ denote that any red-blue coloring of $E(H)$ yields a red copy of $H_{1}$ or a blue copy of $H_{2}$. The Ramsey number for $H_{1}$ versus $H_{2}$, $r(H_{1}, H_{2})$, is the minimum integer $N$ such that $K_{N}\rightarrow (H_{1}, H_{2})$. The Ramsey critical graph $H$ for $H_{1}$ versus $H_{2}$ is a red-blue edge-colored $K_{N- 1}$ such that $H\not\rightarrow (H_{1}, H_{2})$, where $N= r(H_{1}, H_{2})$. In this paper, we characterize all Ramsey critical graphs for a large tree versus $tK_{m}$. As a corollary, we determine the star-critical Ramsey number for a large tree versus $tK_{m}$. |
| title | All Ramsey critical graphs for a large tree versus $tK_{m}$ |
| topic | Combinatorics 05C55, 05D10 |
| url | https://arxiv.org/abs/2506.18235 |