## 2.2 Eulerian Trails and Circuits

Note: if you cannot remember the features of walks, trails and circuits, revise notes for 2.1 Introduction to Walks.

### Eulerian Trails

- A
**Eulerian trail**is a trail which encompasses**every edge**of a graph. - A Eulerian trail will exist if the graph:
- Is connected.
- Has
**exactly two vertices**with an**odd**degree. - Eulerian trails are useful in situations where every edge
**must**be visited, for example when planning a mail route. - A Eulerian trail will start from one of the
**odd**vertices.

Example

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