Editing Cycle space (section)

{{Short description|All even-degree subgraphs of a graph}}
{{about|a concept in [[graph theory]]|space allocated to bicycles|cycling infrastructure}}
In [[graph theory]], a branch of [[mathematics]], the (binary) '''cycle space''' of an [[undirected graph]] is the set of its [[Eulerian graph|even-degree subgraphs]].

This set of subgraphs can be described [[algebra]]ically as a [[vector space]] over the two-element [[finite field]]. The [[dimension (vector space)|dimension]] of this space is the [[circuit rank]] of the graph. The same space can also be described in terms from [[algebraic topology]] as the first [[Homology (mathematics)|homology group]] of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary [[ring (mathematics)|ring]]s.