  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.
