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.


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