  The $k$-edge-colored bipartite Gallai-Ramsey number
$\operatorname{bgr}_k(G:H)$ is defined as the minimum integer $n$ such that
$n^2\geq k$ and for every $N\geq n$, every edge-coloring (using all $k$ colors)
of complete bipartite graph $K_{N,N}$ contains a rainbow copy of $G$ or a
monochromatic copy of $H$. In this paper, we first study the structural theorem
on the complete bipartite graph $K_{n,n}$ with no rainbow copy of $K_{1,3}$.
Next, we utilize the results to prove the exact values of
$\operatorname{bgr}_{k}(P_4: H)$, $\operatorname{bgr}_{k}(P_5: H)$,
$\operatorname{bgr}_{k}(K_{1,3}: H)$, where $H$ is a various union of cycles
and paths and stars.
