TITLE: What are the most important results in graph theory?
QUESTION [8 upvotes]: What are the theorems/results/widely applicable results in graph theory that everyone should know about?

REPLY [3 votes]: Kuratowski's and Wagner's theorems which give necessary and sufficient conditions for a finite graph to be planar.
I should add that it's certainly possible to argue that these are actually theorems of topology rather than graph theory.
For a simple proof of the non-planarity of $K_{3,3}$ and $K_5$ one can consult Munkres's Topology 2nd ed. §64 and at the end of the section he notes "It is a remarkable theorem, due to Kuratowski, that the converse is also true! The proof is not easy."