Is every connected, finite, undirected graph in which all vertices are degree two a cycle?
I feel like this is true, and this is my attempt at a proof:
Start at any vertex. From this vertex, leave to an adjacent vertex, and from there leave to the next and so on. Repeat this process, and every time you get to a vertex, you'll have exhausted one of its outgoing edges, so you will only have one choice -- to move on to the next vertex. Since the graph is finite and connected, eventually you will exhaust all unique vertices and since there is still one vertex (the first vertex we start at) with one unused edge, we must return to it.