Cycle space
In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its Eulerian subgraphs.
This set of subgraphs can be described algebraically as a vector space over the two-element finite field. The 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 group of the graph. Using homology theory, the binary cycle space may be generalized to cycle spaces over arbitrary rings.
A 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 vertices as G itself (this is the meaning of the word "spanning") but, possibly, fewer edges. Thus, a graph G with m edges has 2m 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.
A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has an even number of incident edges (this number is called the 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, 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. The cycle space of a graph is the collection of its Eulerian spanning subgraphs.
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