Graph with multiple Eulerian trails/circuits. Can a graph _g_ that satisfies the necessary condition for having an Eulerian trail have two or more distinct Eulerian trails given a fixed starting vertex? Or, if a graph has an Eulerian trail, is that the only Eulerian trail in that graph?

Note that if you find an Eulerian closed trail, you can also traverse it in opposite direction.

Ignoring this, (you consider the backwards trail the same), it is very easy to prove that a simple Eulerian graph has exactly one trail if and only if it is a cycle.

The reason being that if any vertex has degree $\geq 4$, the trail visits the vertex at least twice. Let $W_1$ be the closed walk between the first and second visit and $W_2$ be the closed walk between the third and 4th visit. Then you can simply interchange them and get a different trail...