[JohnKemeny]

Impressive deduction from someone not into graph theory, but what you're saying is actually known as the Hadwiger conjecture. It says that if a graph isn't t-colorable, then it must have K_t as a minor, where the concept of minor is what you mean (perhaps) with "simplified to".

It so happens that planar graphs are K_5 minor free.

This is touching in on extremely deep theory in the graph theory field.