Moreover, a planar graph’s dual is also
a planar graph, see [5, p. 73]. Thus, to prove the Four Color Theorem, it is equivalent
to prove that no planar graph requires more than four colors to color its vertices. For
the sake of simplicity, from now on, we will refer to a graph as being k-colorable if its
vertices can be colored with k colors.
Let us take a small detour, and prove that K5 is not planar.