TITLE: If a graph can be colored with max $4$ colors, is it planar?
QUESTION [6 upvotes]: There's a theorem that every planar graph can be colored with $4$ colors in such a way that no $2$ adjacent vertices have the same color. Is the opposite true as well?

REPLY [20 votes]: No. Consider $K_{3,3}$, the graph with two sets of 3 vertices each such that every vertex in one set is connected to every vertex in the other. It's not planar but can be colored with just 2 colors. More generally, take any dense bipartite graph - it's still 2-colorable, but far from planar.
A picture of $K_{3,3}$ (along with $K_5$):