Math 2.0

Combinatorics Trees, paths, cycles, connectivity Theorem

Euler's theorem on Eulerian circuits

Builds on Degree of a vertexConnected graphEulerian trail

Statement

Let GG be a finite connected graph or multigraph with at least one edge. Then GG has an Eulerian circuit if and only if every vertex of GG has even degree.

Remarks

The condition is necessary by the theorem on degrees along an Eulerian trail. A consequence: a finite connected graph has an Eulerian trail that is not closed if and only if exactly two vertices have odd degree. Hierholzer's algorithm builds an Eulerian circuit: it joins closed trails until every edge is used.

History

Leonhard Euler showed the necessity of the condition, and he stated without proof that it is sufficient. The first complete proof is by Carl Hierholzer. It was published after his death, in 1873.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionNeighbourhood, definition, no evidence levelNeighbourhooddefinitionWalk, definition, no evidence levelWalkdefinitionDegree of a vertex, definition, no evidence levelDegree of a vertexdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected graph, definition, no evidence levelConnected graphdefinitionEulerian trail, definition, no evidence levelEulerian traildefinitionEuler's theorem on Eulerian circuits, statement, no evidence levelEuler's theorem on Eu…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Neighbourhood definition
  3. Walk definition
  4. Degree of a vertex definition
  5. Reachability definition
  6. Trail definition
  7. Connected graph definition
  8. Eulerian trail definition
  9. Euler's theorem on Eulerian circuits statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Eulerian_path, lead section
About this page
Tag020422MDQF
URIm2://020422MDQF
Kindstatement, theorem
WritingW0 stub
MSC05C45
StatusThis copy of the site has no build data. Run m2 status 020422MDQF.
UsesDegree of a vertex, Connected graph, Eulerian trail

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