What is an Eulerian circuit?

An Eulerian circuit is specifically defined as an Eulerian trail that begins and ends at the same vertex. The key characteristic of an Eulerian circuit is that it must traverse every edge of a graph exactly once while returning to the starting vertex. This property differentiates it from other types of paths or trails in graph theory, making the distinction between an Eulerian circuit and an Eulerian trail (which does not require the starting and ending points to be the same) crucial.

For a graph to have an Eulerian circuit, it must satisfy certain conditions: all vertices must have an even degree, which ensures that the circuit can traverse each edge without getting "stuck" at a vertex. Understanding this situation helps clarify why an Eulerian circuit is a special case of an Eulerian trail, making the statement that it starts and ends at the same vertex essential to its definition.