TITLE: Solving graph theory proofs
QUESTION [9 upvotes]: I am trying to study for an exam on graph theory and I have a few questions. How would you start a proof? For example, when I see a problem like this:

Let G be a graph with n vertices where every vertex has degree at least n/2. Prove that G is
  connected.

How would I start this off? I know how to solve this now but for problems like that, how would I start the question? And for things like, prove that a graph has a bridge, etc. Like, how do I know what to start looking for when I see a problem like that?

REPLY [3 votes]: I will add some tips that I think are helpful when solving graph theory proofs, especially on exams.

Bring a big eraser to exams, as proof writing (especially in graph theory, I have found), involves a lot of trial and error.
First, try a few examples in which the theorem holds and try to think of counterexamples.  Make sure you truly understand what the theorem is stating. 
Take note of why a proposed counterexample is not actually a counterexample.  Often times trying to prove something involves dancing between proposed counterexamples and reasons why they are not counterexamples.  Take note of patterns.
Suppose not (work toward a contradiction).
Consider the contrapositive.
If you are thinking about trying induction, first think about what element (what vertex, if you are inducting on vertices) you will remove from the k+1 graph to get to a valid k graph.  Then think about how you will apply the IH and work forward to the k+1 graph to complete the inductive step.  Don't start writing the structure of the inductive proof until you have thought about these two steps and have an idea about how they will work.  The base case is usually trivial.
Look at biggest or smallest structures, as suggested above (you can always prove the existence of a smallest element, if the set of sizes are a nonempty subset of $\mathbb{N}$ [by WOP], and a maximal element, if the set of sizes are a finite nonempty subset of $\mathbb{N}$).  These structures could be paths, cycles, connected components, etc.
Division into cases
As mentioned above, pigeonhole principle (what are the pigeons and what are the holes?), generalized PHP, handshaking lemma, |V|=|E|-|CC|, consider cycles (or lack thereof), connectivity (or lack thereof), etc.  Generally, know as many theorems as possible... the more you know, the more tools you will have to solve a problem.
Consider the complement of the graph you are working on.
If your theorem says something about "every vertex" or "every edge" or "every connected component," think about one of those elements and what must be true about them
Try to simplify your graph (like reducing the strongly connected components of a digraph into vertices, or reducing structures like cycles into vertices if you wanted to prove something about cycle adjacency, for instance)

Practicing is the best way to get better at this kind of thing.