Why should it be that a doodle is always two-colorable? It's a question a child could ask, but it leads to the whole area of graph theory. Personally, I find it a much better introduction to Graph Theory than the usual question about the Bridges of Königsberg:

https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg

But there is a progression.

* A doodle is two-colorable

* A doodle is a planar, connected, Eulerian graph

* The dual of a planar, connected, Eulerian graph is bi-partite

* Bi-partite graphs are trivial to identify, and trivial to colour.

* What about tri-partite graphs?

* Identifying whether a given graph is tri-partite is NP-Complete.