Gallai-Ramsey Numbers for Rainbow Paths

Given graphs G and H and a positive integer k, the 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 is an element of {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.