Some comments from Jose

1. When you are writing a recurrence relation it is standard to write the n-th term (or the (n+1)-st term) as a function of the previous terms. You should also provide initial conditions.
2. You should never say that there is a very clear bijection between two sets and not show how to construct it. This was a very common mistake in problem #2.
3. If you write a few first terms of a sequence and find a recurrence relation that these few terms satisfy, you then should also prove that the entire sequence really satisfies it. One standard way to do this is by induction.
4. If you write in your solution that a certain object exists you should construct it, or indicate how to construct it. Many of you claimed that the number of odd degree vertices being even is enough to be able to construct a graph with a given degree sequence. However the sequence in problem #4 is a counterexample to that claim.
5. In the last problem you should NOT say anything like: "assume the graph looks like this". This usually does NOT cover all the cases you have to cover to solve the problem.
6. One thing you should have learned from this homework: The trick of adding up all the degrees is usually a great idea and an approach that should always be considered, but it neither solves all the problems nor entirely characterizes the possible degree sequences of graphs.