A first lecture in graph theory

Here I look at a potential first lecture in a second year graph theory course. I believe that this type of lecture can provide an interesting introduction to a new topic that engages most of the students. It also provides feedback to the lecturer on the students' level of ability. But more than that, it recognizes that we all need to play with ideas to make them part of our knowledge pool and to increase our learning and understanding. Traditional lectures require students to sort out the basics of the subject for themselves. I believe that students don't do this well and hence need to be taught how to ‘play’ mathematically.