Gallai-Ramsey Numbers for Rainbow $S_3^+$ and Monochromatic Paths

Motivated by Ramsey theory and other rainbow-coloring-related problems, we consider edge-colorings of complete graphs without rainbow copy of some fixed subgraphs. Given two graphs $G$ and $H$, the $k$-colored Gallai-Ramsey number $ gr_k(G : H)$ is defined to be the minimum positive integer $n$ such that every $k$-coloring of the complete graph on $n$ vertices contains either a rainbow copy of $G$ or a monochromatic copy of $H$. Let $ S_3^+$ be the graph on four vertices consisting of a triangle with a pendant edge. In this paper, we prove that $ gr_k(S_3^+ : P_5) = k+4 (k \ge 5)$, $ gr_k(S_3^+ : mP_2) = (m-1)k+m+1 (k \ge 1) $, $ gr_k(S_3^+ : P_3 \cup P_2) = k+4 (k \ge 5) $ and $ gr_k( S_3^+ : 2P_3) = k+5 (k \ge1) $.