A video game world:
A surface is a space that "locally looks like" $\mathbb{R}^2$
Answer: After break
The complete bipartite graph $K_{n,m}$ has $n$ red vertices, $m$ blue vertices, and every red vertex is connected to every blue vertex.
A graph is planar if it can be drawn on a sheet of paper without any edges crossing
So the utilities question can be rephrased as: is $K_{3,3}$ planar?
Need a way to organize the cases...
Every simple closed curve separates the plane into an inside and an outside
If $\Gamma$ can be drawn in the plane, any cycle has to make a loop; every other edge must be either inside or outside.
Want as many edges as possible in the loop. Best case: $G$ is Hamiltonian
Eventually, this leads to the "Planarity Algorithm", but first, $K_{3,3}$ and $K_5$...