  Given two graphs $G$ and $H$, the \emph{general $k$-colored Gallai-Ramsey
number} $\operatorname{gr}_k(G:H)$ is defined to be the minimum integer $m$
such that every $k$-coloring of the complete graph on $m$ vertices contains
either a rainbow copy of $G$ or a monochromatic copy of $H$. Interesting
problems arise when one asks how many such rainbow copy of $G$ and
monochromatic copy of $H$ must occur. The \emph{Gallai-Ramsey multiplicity}
$\operatorname{GM}_{k}(G,H)$ is defined as the minimum total number of rainbow
copy of $G$ and monochromatic copy of $H$ in any exact $k$-coloring of
$K_{\operatorname{gr}_{k}(G,H)}$. In this paper, we give upper and lower bounds
for Gallai-Ramsey multiplicity involving some small rainbow subgraphs.
