Write out solutions to all the questions you do, not only the ones for handing in.
Do as many questions as you can that are not hand-in problems. The class discussion is about them, and discussion is how you learn half your graph theory. If you haven't tried the problems, you don't get so much out of the class discussion, nor out of talking with your friends.
Besides finding the answer, always try to explain, as well as you can, how you know you have the correct answer. This is an important part of your solution. An answer without explanation is not worth much (except for very simple questions).
When solving problems, a systematic solution is better than guesswork. You often may find a solution by intelligent guessing, but then you should look for a way of showing that your solution is correct. This part needs to be systematic if it is to be completely convincing. (This will be clearer after a few days of class!)
Allow 15 minutes per problem (minimum) before you give up, even if you feel you're getting nowhere. These problems need time for thought. If you're still stuck, go on to another problem. Return to the sticky problem later (say, the next day). Often, it then looks easier because you tried hard the first time and then gave your mind time to grind it up--I mean, to come up with ideas. To get the advantage of this method, you have to start the problems well ahead of time. Last-minute effort will not work well in this class.