What are Eulerian Circuits and Trails? [Graph Theory]

What are Eulerian circuits and trails? This video explains the definitions of eulerian circuits and trails, and provides examples of both and their interesting properties.

Eulerian trails are a trail in a graph (that is, a walk without repeating edges) that uses each edge in the graph exactly once. That means the trail covers all the edges of the graph, without repeating any of those edges. Eulerian circuits are a special kind of eulerian trail that starts and ends on the same vertex. Note that both Eulerian trails and circuits may contain repeated vertices, but repeated edges are forbidden in both. Sometimes, Eulerian trails are called Eulerian paths, Euler tours, or Eulerian tours. Also, some refer to Eulerian circuits as an Eulerian cycle, although generally a cycle refers to circuits with the added restriction that there are no repeated vertices.

Eulerian trails and circuits and relatively easy to find in a graph, at least when compared to Hamiltonian paths and cycles. Some of their interesting properties include: all connected graphs where every vertex is of even degree contains an Eulerian circuit; if a graph contains more than 2 odd-degree vertices, there can be no Eulerian trail or circuit in that graph; the line graph of a connected graph with an Eulerian circuit or trail will have a Hamiltonian cycle or path. Eulerian trails find applications in bioinformatics and circuit design, among other fields.

(Here's the interactive mentioned in the video):
https://d3gt.com/unit.html?eulerian-t...

For more information, visit these links:
https://en.wikipedia.org/wiki/Euleria...
https://www.geeksforgeeks.org/euleria...

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