  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 consider two classes of
unicyclic graphs, the star with an extra edge and the path with a triangle at
one end. We provide the $2$-color Ramsey numbers for these two classes of
graphs and use these to obtain general upper and lower bounds on the
Gallai-Ramsey numbers.
