Question: 1. Conjecture a necessary condition for a graph to have an Euler circuit (i.e., if a graph has an Euler circuit, this condition must hold).
2. Make a conjecture about what property (or properties) a graph needs to have (or not have) for it to be guaranteed to have an Euler circuit.
3. Make similar conjectures about Euler traversals.
4. Try to prove your conjectures.