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.