Not properly worded, but we want to find the minimum possible $k$, such that we always can chose $k$ colors and after removing the rest ones, the graph is still connected. $k=7$. The idea is that for any graph $G$, either $G$ or its complement $\overline{G}$ is connected. The example that we cannot do better than $7$ colors is more interesting. See St. Petersburg 2016 , 9.6