The problem is:
Let $G$ be a connected planar graph with less than 12 vertices. a) Prove that G has a vertex with degree $\leq 4$. b) Prove that $\chi (G) \leq 4$. (Do not use the Four Color Theorem)
I did prove part a) satisfactorily I think at least =/
My question is about part b).
I am thinking showing part b) by contradiction may work. I am thinking perhaps something concerning a particular vertex with degree at most $4$, deleting this vertex, showing a $4$ coloring is still possibly, any hints? =/
I am also thinking I could use Euler's Formula but I am not sure of how I would relate $n+r-e=2$ back to the chromatic number.
(I should have stated it said to do the problem without the four color theorem)