  Given two non-empty graphs $G,H$ and a positive integer $k$, the
Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum
integer $N$ such that for all $n\geq N$, every $k$-edge-coloring of $K_n$
contains either a rainbow colored copy of $G$ or a monochromatic copy of $H$.
In this paper, we got some exact values or bounds for
$\operatorname{gr}_k(P_5:H) \ (k\geq 3)$ if $H$ is a general graph or a star
with extra independent edges or a pineapple.
