Eulerian path

Eulerian path is a notion from graph theory. A eulerian path in a graph is one that visits each edge of the graph once only. A Eulerian circuit or Eulerian cycle is an Eulerian path which starts and ends on the same vertex.

Eulerian Path Media

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  Multigraphs of both Königsberg Bridges and Five room puzzles have more than two odd vertices (in orange), thus are not Eulerian and hence the puzzles have no solutions.

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  Every vertex of this graph has an even degree. Therefore, this is an Eulerian graph. Following the edges in alphabetical order gives an Eulerian circuit/cycle.

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  An infinite graph with all vertex degrees equal to four but with no Eulerian line

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  A directed graph with all even degrees that is not Eulerian, serving as a counterexample to the statement that a sufficient condition for a directed graph to be Eulerian is that it has all even degrees

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  This mixed graph is Eulerian. The graph is even but not symmetric which proves that evenness and symmetricness are not necessary and sufficient conditions for a mixed graph to be Eulerian.

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  An even mixed graph that violates the balanced set condition and is therefore not Eulerian.

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  An even mixed graph that satisfies the balanced set condition and is therefore an Eulerian mixed graph.