Editing Eulerian path (section)

==In infinite graphs==
[[File:Kely graph of F2 clear.svg|thumb|An infinite graph with all vertex degrees equal to four but with no Eulerian line]]
In an [[infinite graph]], the corresponding concept to an Eulerian trail or Eulerian cycle is an Eulerian line, a doubly-infinite trail that covers all of the edges of the graph. It is not sufficient for the existence of such a trail that the graph be connected and that all vertex degrees be even; for instance, the infinite [[Cayley graph]] shown, with all vertex degrees equal to four, has no Eulerian line. The infinite graphs that contain Eulerian lines were characterized by {{harvtxt|Erdős|Grünwald|Weiszfeld|1936}}. For an infinite graph or multigraph {{mvar|G}} to have an Eulerian line, it is necessary and sufficient that all of the following conditions be met:<ref>{{citation
 | last = Komjáth | first = Peter | author-link = Péter Komjáth
 | contribution = Erdős's work on infinite graphs
 | contribution-url = https://books.google.com/books?id=7_zFBAAAQBAJ&pg=PA325
 | doi = 10.1007/978-3-642-39286-3_11
 | mr = 3203602
 | pages = 325–345
 | publisher = János Bolyai Math. Soc., Budapest
 | series = Bolyai Soc. Math. Stud.
 | title = Erdös centennial
 | volume = 25
 | year = 2013}}.</ref><ref>{{citation
 | last = Bollobás | first = Béla | author-link = Béla Bollobás
 | doi = 10.1007/978-1-4612-0619-4
 | isbn = 0-387-98488-7
 | mr = 1633290
 | page = 20
 | publisher = Springer-Verlag, New York
 | series = Graduate Texts in Mathematics
 | title = Modern graph theory
 | url = https://books.google.com/books?id=JeIlBQAAQBAJ&pg=PA20
 | volume = 184
 | year = 1998}}.</ref>
*{{mvar|G}} is connected.
*{{mvar|G}} has [[countable set]]s of vertices and edges.
*{{mvar|G}} has no vertices of (finite) odd degree.
*Removing any finite subgraph {{mvar|S}} from {{mvar|G}} leaves at most two infinite connected components in the remaining graph, and if {{mvar|S}} has even degree at each of its vertices then removing {{mvar|S}} leaves exactly one infinite connected component.