Gallai-Ramsey Numbers for $\mathrm{\ell }$-Connected Graphs

Given a nonempty graph $G$, a collection of nonempty graphs $\mathcal{H}$, and a positive integer $k$, the Gallai-Ramsey number ${\mathrm{g}\mathrm{r}}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact $k$-edge-coloring of a complete graph $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of some element in $\mathcal{H}$. In this paper, we obtain some exact values and general lower and upper bounds for ${\mathrm{g}\mathrm{r}}_k(G:{\mathcal{F}}^{\mathrm{\ell }})$, where ${\mathcal{F}}^{\mathrm{\ell }}$ is the set of $\mathrm{\ell }$-connected graphs and $G\in \{P_5,K_{1,3}\}$.