On some path-critical Ramsey numbers
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916146428510208 |
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| author | Wang, Ye Song, Yanyan |
| author_facet | Wang, Ye Song, Yanyan |
| contents | For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The path-critical Ramsey number $R_π(G,H)$ is the largest $n$ such that any red-blue edge coloring of $K_r \setminus P_{n}$ contains a red $G$ or a blue $H$, where $r=R(G,H)$ and $P_{n}$ is a path of order $n$. In this note, we show a general upper bound for $R_π(G,H)$, and determine the exact values for some cases of $R_π(G,H)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_02641 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On some path-critical Ramsey numbers Wang, Ye Song, Yanyan Combinatorics For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $r$ such that any red-blue edge coloring of $K_r$ contains a red $G$ or a blue $H$. The path-critical Ramsey number $R_π(G,H)$ is the largest $n$ such that any red-blue edge coloring of $K_r \setminus P_{n}$ contains a red $G$ or a blue $H$, where $r=R(G,H)$ and $P_{n}$ is a path of order $n$. In this note, we show a general upper bound for $R_π(G,H)$, and determine the exact values for some cases of $R_π(G,H)$. |
| title | On some path-critical Ramsey numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.02641 |