What condition must hold for an Eulerian circuit to exist?

For an Eulerian circuit to exist, it is essential that all vertices in the graph have an even degree. This is because, in an Eulerian circuit, each time you enter a vertex, you must also exit it. If a vertex has an odd degree, there would be an unbalanced entry and exit count, making it impossible to return to that vertex as part of a continuous circuit.

In graphs that meet this criterion, every vertex can be traversed without lifting the pen and returning to the starting point, which characterizes an Eulerian circuit. Therefore, the requirement for all vertices to have even degrees ensures that the circuit is completed seamlessly without any interruptions.

While other conditions relate to different types of Eulerian paths or circuits, they do not meet the criteria necessary for an Eulerian circuit specifically.