Gallai–Ramsey Numbers for Paths

For two graphs G and H, the Gallai–Ramsey number \({\text {gr}}_k(G:H)\) ${\text{gr}}_k(G:H)$ is defined as the minimum integer n such that any k-edge-coloring of \(K_n\) $K_n$ must contain either a rainbow copy of G or a monochromatic copy of H. In this paper, we obtain the exact values of \({\text {gr}}_k(G:H)\) ${\text{gr}}_k(G:H)$ , where H is a path and \(G\in \{K_{1,3},P_4^+,P_5\}\) $G\in \{K_{1,3},P_4^{+},P_5\}$ is a small tree and \(P_4^+\) $P_4^{+}$ is the graph consisting of \(P_4\) $P_4$ with one extra edge incident with an inner vertex.