Abstract
For decades, Desert Island Discs has been a popular radio program on the BBC. Its format is for a guest to choose a selection of musical recordings that they would like to have with them if they were stranded on a desert island. In this chapter, we will unashamedly imitate that program and select some conjectures that we would like to have with us were we to be stranded on a desert island. Our selections go back about half a century, to when we were first introduced to graph theory. In the first part of this chapter, our conjectures involve drawings and embeddings of graphs on surfaces, but in the second part, we present one on connectivity.
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