Gallai-Ramsey numbers for fans

Given a graph $G$ and a positive integer $k$, define the \emph{Gallai-Ramsey number} to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for fans $F_m=K_1+m{K}_2$ and prove the sharp result for $m=2$ and for $m=3$ with $k$ even.