Eulerian Graph
An Eulerian graph is a graph containing an Eulerian cycle. The numbers of Eulerian graphs with 
, 2, ... nodes
are 1, 1, 2, 3, 7, 15, 52, 236, ... (OEIS A133736),
the first few of which are illustrated above.
The corresponding numbers of connected Eulerian graphs are 1, 0, 1, 1, 4, 8, 37, 184, 1782, ... (OEIS A003049; Robinson 1969; Liskovec 1972; Harary and Palmer 1973, p. 117), the first few of which are illustrated above.
Some care is needed in interpreting the term, however, since some authors define an Euler graph as a different object, namely a graph
for which all vertices are of even degree (motivated by the following theorem). Euler
showed (without proof) that a connected simple
graph is Eulerian iff it has no graph
vertices of odd degree
(i.e., all vertices are of even degree). While the number of connected Euler graphs
on 
nodes is equal to the number of connected Eulerian
graphs on 
nodes, the counts are different for disconnected
graphs since there exist disconnected graphs having multiple disjoint cycles with
each node even but for which no single cycle passes through all edges.
A directed graph is Eulerian iff every graph vertex has equal indegree
and outdegree. A planar bipartite
graph is dual to a planar
Eulerian graph and vice versa. The numbers of Eulerian digraphs on 
, 2, ... nodes
are 1, 1, 3, 12, 90, 2162, ... (OEIS A058337).
Finding the largest subgraph of graph having an odd number of vertices which is Eulerian is an NP-complete problem (Skiena 1990, p. 194).
A graph can be tested in the Wolfram Language to see if it Eulerian using the command EulerianGraphQ[g].
The following table gives some named Eulerian graphs.
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