1. Ramsey and Gallai-Ramsey numbers for linear forests and kipas
- Author
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Li, Ping, Mao, Yaping, Schiermeyer, Ingo, and Yao, Yifan
- Subjects
Mathematics - Combinatorics - Abstract
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.
- Published
- 2024