Editing Eulerian path (section)

== Mixed Eulerian graphs ==
[[File:Eulerian mixed graph that is even but not symmetric proving that evenness and symmetricness is not a necessary and sufficient condition for a mixed graph to be Eulerian.svg|alt=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.|thumb|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.]]
All [[Mixed graph|mixed graphs]] that are both even and symmetric are guaranteed to be Eulerian. However, this is not a necessary condition, as it is possible to construct a non-symmetric, even graph that is Eulerian.<ref name=":0"/>

[[L._R._Ford_Jr.|Ford]] and [[D._R._Fulkerson|Fulkerson]] proved in 1962 in their book ''Flows in Networks''<ref>{{cite book
 | author = L. R. Ford 
 |author2=D. R. Fulkerson
 | year = 1962
 | title = Flows in Networks
 | url = https://archive.org/details/flowsinnetworks0000ford
 | url-access = registration
 | publisher = Princeton University Press
 | location = Princeton, NJ
|isbn=9780691079622
 }}</ref> 
a necessary and sufficient condition for a graph to be Eulerian, viz., that every vertex must be even and satisfy the balance condition, i.e. for every subset of vertices S, the difference between the number of arcs leaving S and entering S must be less than or equal to the number of edges incident with S.<ref name=":0" />

The process of checking if a mixed graph is Eulerian is harder than checking if an undirected or directed graph is Eulerian because the balanced set condition concerns every possible subset of vertices.