EulerianGraphQ

EulerianGraphQ works with undirected graphs:

Directed graphs:

EulerianGraphQ gives False for expressions which are not graphs:

Works with large graphs:

Euler's "Seven Bridges of Köningsberg", with multiple edges removed:

Test whether the graph is Eulerian:

Label the edges visited along the Eulerian cycle:

Find an Eulerian cycle:

An Eulerian cycle can be found using FindEulerianCycle:

A connected undirected graph is Eulerian iff every graph vertex has an even degree:

A connected undirected graph is Eulerian if it can be decomposed into edge disjoint cycles:

The graphs are cycles if they are connected and have equal number of edges and vertices:

For connected directed graphs:

The line graph of an undirected Eulerian graph is Eulerian:

The line graph of an Eulerian graph is Hamiltonian:

A connected directed graph is Eulerian iff every vertex has equal in-degree and out-degree:

Cycle graphs are Eulerian: