Ramsey and Gallai-Ramsey numbers for linear forests and kipas

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} ${\mathrm{gr}}_k(G:H)$ 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 ${\mathrm{gr}}_k(G:H)$, 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.