Given graphs
G and
H and a positive integer
k, the \emph{Gallai-Ramsey number}, denoted by
grk(G:H) is defined to be the minimum integer
n such that every coloring of
Kn 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∈{P4,P5}. In the case where
G=P4, we completely solve the Gallai-Ramsey question by reducing to the
2-color Ramsey numbers. In the case where
G=P5, we conjecture that the problem reduces to the
3-color Ramsey numbers and provide several results in support of this conjecture.