It seems that most current texts agree that an Eulerian graph is one in which there is a closed walk, no edge can be repeated. In this case show that all vertices are even.
On the other hand, we can find older texts that talk about traceable graphs being Eulerian. That is, there is a walk( not necessarily closed) that includes each edge exactly once. In this case, there are at most two odd vertices.
So check your text material for the definition in use.