Editing Cycle space (section)

===Graph theory===
A [[Glossary of graph theory#Subgraphs|spanning subgraph]] of a given graph ''G'' may be defined from any subset ''S'' of the edges of ''G''. The subgraph has the same set of [[Vertex (graph theory)|vertices]] as ''G'' itself (this is the meaning of the word "spanning") but has the elements of ''S'' as its edges. Thus, a graph ''G'' with ''m'' edges has 2<sup>''m''</sup> spanning subgraphs, including ''G'' itself as well as the [[empty graph]] on the same set of vertices as ''G''. The collection of all spanning subgraphs of a graph ''G'' forms the [[edge space]] of ''G''.<ref name="gy">{{citation|title=Graph Theory and Its Applications|edition=2nd|first1=Jonathan L.|last1=Gross|first2=Jay|last2=Yellen|publisher=CRC Press|year=2005|isbn=9781584885054|chapter=4.6 Graphs and Vector Spaces|pages=197–207|url=https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA197}}.</ref><ref name="diestel">{{citation|title=Graph Theory|volume=173|series=Graduate Texts in Mathematics|first=Reinhard|last=Diestel|publisher=Springer|year=2012|chapter=1.9 Some linear algebra|pages=23–28|url=https://books.google.com/books?id=eZi8AAAAQBAJ&pg=PA23}}.</ref>

A graph ''G'', or one of its subgraphs, is said to be [[Eulerian graph|Eulerian]] if each of its vertices has an [[even number]] of incident edges (this number is called the [[degree (graph theory)|degree]] of the vertex). This property is named after [[Leonhard Euler]] who proved in 1736, in his work on the [[Seven Bridges of Königsberg]], that a [[connected graph]] has a tour that visits each edge exactly once if and only if it is Eulerian. However, for the purposes of defining cycle spaces, an Eulerian subgraph does not need to be connected; for instance, the empty graph, in which all vertices are disconnected from each other, is Eulerian in this sense.  The cycle space of a graph is the collection of its Eulerian spanning subgraphs.<ref name="gy"/><ref name="diestel"/>