An Eulerian cycle, also called an Eulerian circuit, Euler circuit, Eulerian tour, or Euler tour, is a closed trail that uses every graph edge exactly once, starting and ending at the same graph vertex. For technical reasons, Eulerian cycles are mathematically easier to study than are Hamiltonian cycles. An Eulerian cycle for the octahedral graph is illustrated above.
As a generalization of the Königsberg bridge problem, Euler showed (without proof) that a connected graph has an Eulerian cycle iff it has no graph vertices of odd vertex degree.
Fleury's algorithm is an elegant, but inefficient, method of generating an Eulerian cycle. An Eulerian cycle of a graph may be found in the Wolfram Language using FindEulerianCycle[g].
For directed Eulerian graphs, the number of Eulerian cycles can be counted using the BEST theorem.
The only Platonic solid whose skeleton possesses an Eulerian cycle is the octahedron, which
has Schläfli symbol 
; all other Platonic graphs
have odd vertex degrees.
Similarly, the only Eulerian Archimedean solids
are the cuboctahedron, icosidodecahedron,
small rhombicosidodecahedron, and
small rhombicuboctahedron.
