Ramsey and Gallai-Ramsey numbers for linear forests and kipas
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| Format: | Preprint |
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2024
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| _version_ | 1866913208563924992 |
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| author | Li, Ping Mao, Yaping Schiermeyer, Ingo Yao, Yifan |
| author_facet | Li, Ping Mao, Yaping Schiermeyer, Ingo Yao, Yifan |
| contents | For two graphs $G,H$, the \emph{Ramsey number} $r(G,H)$ is the minimum integer $n$ such that any red/blue edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. For two graphs $G,H$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $n$ such that any $k$-edge-coloring of $K_n$ must contain either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, the classical Ramsey numbers of linear forest versus kipas are obtained. We obtain the exact values of $\operatorname{gr}_k(G:H)$, where $H$ is either a path or a kipas and $G\in\{K_{1,3},P_4^+,P_5\}$ and $P_4^+$ is the graph consisting of $P_4$ with one extra edge incident with inner vertex. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_08942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramsey and Gallai-Ramsey numbers for linear forests and kipas Li, Ping Mao, Yaping Schiermeyer, Ingo Yao, Yifan Combinatorics For two graphs $G,H$, the \emph{Ramsey number} $r(G,H)$ is the minimum integer $n$ such that any red/blue edge-coloring of $K_n$ contains either a red copy of $G$ or a blue copy of $H$. For two graphs $G,H$, the \emph{Gallai-Ramsey number} $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $n$ such that any $k$-edge-coloring of $K_n$ must contain either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, the classical Ramsey numbers of linear forest versus kipas are obtained. We obtain the exact values of $\operatorname{gr}_k(G:H)$, where $H$ is either a path or a kipas and $G\in\{K_{1,3},P_4^+,P_5\}$ and $P_4^+$ is the graph consisting of $P_4$ with one extra edge incident with inner vertex. |
| title | Ramsey and Gallai-Ramsey numbers for linear forests and kipas |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.08942 |