What is an Eulerian trail?

An Eulerian trail is a specific type of path in graph theory that visits every edge of a graph exactly once while allowing for the possibility of visiting vertices multiple times. This is significant because it highlights the key attribute of an Eulerian trail: the focus on the edges rather than the vertices.

For a trail to be considered Eulerian, it must meet certain conditions regarding the degrees of the vertices in the graph. Specifically, an Eulerian trail can exist in a connected graph if there are either zero or exactly two vertices of odd degree. When there are exactly two vertices of odd degree, the trail will start at one of these vertices and end at the other.

Option A accurately describes this concept, emphasizing the visitation of every edge exactly once, which is the defining characteristic of an Eulerian trail. Understanding this concept is essential in graph theory, particularly in applications such as routing problems, network design, and even game design, where the movement through pathways (edges) is of interest.

The other options present different types of trails or circuits. A trail with no repeated edges refers to a different concept where a trail may not revisit any edge but doesn't necessarily cover all edges. A circuit starting and ending at the same vertex is a different structure known as