Gallai–Ramsey Numbers for Rainbow Paths

Abstract

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 $G \in \{P_{4}, P_{5}\}$. 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.