Unicursal Graphs and Unicursal Lines | Semi-Eulerian Trails, Graph Theory

What are unicursal graphs and unicursal lines in graph theory? We'll define them, and provide examples in today's lesson!

A unicursal line of a graph is an open trail containing every edge of the graph. By definition of open trail, it contains each edge exactly once, but may repeat vertices, but must start and end at distinct vertices. A unicursal graph is a graph containing a unicursal line.

Recall that an Eulerian circuit is a circuit containing every edge of a graph. A connected graph contains an Eulerian circuit if and only if all of its vertices have even degree. Proof:    • Proof: Graph is Eulerian iff All Vert...  

Using that theorem we will also explain why a connected graph is unicursal if and only if it has exactly two odd vertices!

Unicursal lines are also sometimes called semi-Eulerian trails.



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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