You're mistaken about that not being 4-colorable. Color the "rings", from outside to inside: red, blue, green, blue, yellow. Again, as mentioned in my earlier

You're mistaken about that not being 4-colorable. Color the "rings", from outside to inside: red, blue, green, blue, yellow. Again, as mentioned in my earlier comment, I would suggest thinking of things / describing things in terms of plane graphs rather than "maps". It'll make everything easier. I might also suggest learning some of the relevant graph theory? It sounds like you're trying and failing to embed a K_5, or perhaps an arbitrary K_n, into the plane. That can't be done, and the proof that this is impossible is much easier than the full 4-color theorem. (There are also the 5-color and 6-color theorems, which again have much easier proofs than the 4-color theorem, and which are another reason you can't embed an arbitrary K_n in the plane.)