Editing Cycle (graph theory) (section)

== Cycle space ==
The term ''cycle'' may also refer to an element of the [[cycle space]] of a graph. There are many cycle spaces, one for each coefficient field or ring. The most common is the ''binary cycle space'' (usually called simply the ''cycle space''), which consists of the edge sets that have even degree at every vertex; it forms a [[vector space]] over the two-element [[finite field|field]]. By [[Veblen's theorem]], every element of the cycle space may be formed as an edge-disjoint union of simple cycles. A [[cycle basis]] of the graph is a set of simple cycles that forms a [[basis (linear algebra)|basis]] of the cycle space.<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|chapter-url=https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA197|access-date=2016-09-27|archive-date=2023-02-04|archive-url=https://web.archive.org/web/20230204155009/https://books.google.com/books?id=-7Q_POGh-2cC&pg=PA197|url-status=live}}.</ref>

Using ideas from [[algebraic topology]], the binary cycle space generalizes to vector spaces or [[module (mathematics)|modules]] over other [[ring (mathematics)|rings]] such as the integers, rational or real numbers, etc.<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|chapter-url=https://books.google.com/books?id=eZi8AAAAQBAJ&pg=PA23|access-date=2016-09-27|archive-date=2023-02-04|archive-url=https://web.archive.org/web/20230204155010/https://books.google.com/books?id=eZi8AAAAQBAJ&pg=PA23|url-status=live}}.</ref>