For any $11$-vertex graph $G$, show that $G$ and $\overline{G}$ cannot both be planar Let $G$ be a graph with 11 vertices. Prove that $G$ or $\overline{G}$ must be nonplanar.
This question was given as extra study material but a little stuck. Any intuitive explanation would be great!

It seems the following. Euler formula implies that $E\le 3V-6$ for each planar graph. If both $G$ and $\bar G$ are planar, then $55=|E(K_{11})|\le 6|V(K_{11})|-12=54$, a contradiction.
