  Given graphs $G$ and $H$ and a positive integer $k$, the \emph{Gallai-Ramsey
number}, denoted by $gr_{k}(G : H)$ is defined to be the minimum integer $n$
such that every coloring of $K_{n}$ using at most $k$ colors will contain
either a rainbow copy of $G$ or a monochromatic copy of $H$. We consider this
question in the cases where $G \in \{P_{4}, P_{5}\}$. In the case where $G =
P_{4}$, we completely solve the Gallai-Ramsey question by reducing to the
$2$-color Ramsey numbers. In the case where $G = P_{5}$, we conjecture that the
problem reduces to the $3$-color Ramsey numbers and provide several results in
support of this conjecture.
