[jmblpati]

Quick addendum: although the four color theorem proof always seems to require a case analysis at some point, proving that planar graphs admit 5-colorings can be done with a very short proof. Not particularly relevant to the analogy in the post, but if you want proof that planar graphs admit constant-sized colorings that's the one for you.

[ocfnash]

Just to advertise this addendum a bit more!

The Kempe-chain proof of the 5-colouring of any planar graph is easy and beautiful.