Graph Theory. Graph theory is the study of graphs, which are tools for studying the interactions between objects at a very abstract level. It is arguably the second-most applicable field of mathematics outside of linear algebra, because its central objects lend themselves well to modeling many kinds of phenomena. It has been gainfully applied in networks, traffic flow, electrical circuits, decision analysis, and game theory. It also is extremely applicable inside of mathematics because of the deep relationship between graphs and certain algebraic objects.
Graph theory proofs tend to be pretty dull to read incorrectly. They come from these amazing pictures, and then they get translated into linear writing. People who work with graphs have (collectively) gotten very good at doing this, and there is a very rich language with which to do it; so rich that it is not even very hard to skip over the pictures entirely. You can read the text and follow its validity, and the proof is over. But the proofs are always much more satisfying if you have a pencil in hand.
I tried to put in a couple of pictures that came up while I was thinking about this proof, but there were many, many more.