If a network has a trail or circuit that uses every edge, we call the network traversable (sometimes semi-Eulerian). This special walk is then called an Euler trail* if it’s open or an Euler circuit if it’s closed, named after the same Euler that gave us Euler’s formula. If a network has an Euler circuit we call it an Euler network (sometimes Eulerian).
Some texts refer to such a trail as an Euler path, but this special walk is a trail (since vertices can be repeated), not a path!
You can think about a traversable networks as ones where you can start at a vertex and draw in all the edges without taking your pen (or finger or stylus) off the page (or screen). Here are two traversable networks, the one on the right is an Euler network:
Each network has a trail that uses every edge exactly once. The trail on the right is closed, so it is a circuit, and any vertex can be the start/end
We will discover a way to find these trails and circuits in a later lesson.