An important subspace of the edge space is the **cycle space**. It consists of
(the edge sets of)
all the cycles of the graph and of all sums of cycles. The addition of two
cycles (shown dashed) is illustrated in the figure.
Note that the result here (also shown dashed) is not a cycle.

There are a number of basic results concerning the cycle space. Firstly, the elements of the cycle space can be characterized by the cuts: A set of edges is an element of the cycle space if and only if it meets every cut in a finite number of edges.

Secondly, the dimension of the cycle space is related to the number of vertices of
and edges of the graph. If the graph has *n* vertices and *m* edges then
the dimension is: dim = *m*-*n*+1.

Thirdly, the cycle space is generated by the fundamental cycles of every spanning tree.

An important application of the cycle space is Mac Lane's planarity criterion, which gives an algebraic characterization of the planar graphs.

**References:**

- R. Diestel,
*Graph Theory*(2nd edition), Springer-Verlag, 2000