Under what condition can an Eulerian trail exist?

An Eulerian trail is a path within a graph that visits every edge exactly once. For a graph to contain an Eulerian trail, a specific condition regarding the vertices must be met.

The correct condition is that there must be exactly two odd vertices in the graph. This configuration allows the trail to start at one of the odd vertices and end at the other. If there are no odd vertices at all, the graph will have an Eulerian circuit instead, meaning that you can traverse every edge and return to the starting point, but that configuration does not allow for a start and end at different vertices.

Conversely, a situation with more than two odd vertices cannot support an Eulerian trail because it would not be possible to pair off all odd degrees with a return to even degrees. The other options suggest scenarios that do not adhere to the fundamental properties of graph theory concerning Eulerian trails, thus failing to satisfy the requirement for their existence.