Cycle space

This article is about a concept in graph theory. For space allocated to bicycles, see segregated cycle facilities.

In graph theory, an area of mathematics, a cycle space is a vector space defined from an undirected graph; elements of the cycle space represent formal combinations of cycles in the graph. Cycle spaces allow one to use the tools of linear algebra to study graphs. A cycle basis is a set of cycles that generates the cycle space.

The binary cycle space

Let G be a finite simple undirected graph with edge set E. The power set of E becomes a Z₂-vector space if we take the symmetric difference as addition, identity function as negation, and empty set as zero. The one-element sets form a basis, so its dimension is equal to the number of edges of G. Because every element of this vector space is a subset of E, it can be regarded as an indicator function of type $E\to \mathbb{Z}_2$, then this vector space coincide with the free Z₂-module with basis E. This is the (binary) edge space of G.

An important subspace of the edge space is the (binary) cycle space. It is by definition the subspace generated by (the edge sets of) all the simple cycles of the graph. The addition of two cycles (shown dashed) is illustrated in the figure. Note that the result here (also shown dashed) is not a simple cycle but an edge-disjoint union of two simple cycles.

There are a number of basic results concerning the cycle space. The symmetric difference of two simple cycles is either a simple cycle or a union of edge-disjoint simple cycles. Using this observation, one can show that an edge set is in the cycle space if and only if it is a disjoint union of simple cycles. Phrased another way: the set F of edges is in the cycle space if and only if every vertex in the subgraph spanned by F has even degree.

It is not necessary to use all cycles to generate the cycle space: if G is connected and any spanning tree T of G is given, then the fundamental cycles of T form a basis of the cycle space, known as a cycle basis. The dimension of the cycle space of a connected graph is thus related to the number of vertices and edges of the graph. If the graph has n vertices and m edges then the dimension is m − n + 1. In a planar graph, the set of interior faces of an embedding of the graph also provides a cycle basis.

An important application of the cycle space are Whitney's planarity criterion and Mac Lane's planarity criterion, which give an algebraic characterization of the planar graphs.

The integral cycle space

The foregoing development can be carried out over the integers, Z. The integral edge space is the abelian group Z^E of functions from the edge set E to the integers. It is necessary (for the notation) to choose an arbitrary orientation of the graph in order to define the cycle space, but the definition does not depend on that choice. An integral cycle is a function such that the sum of values on edges oriented into a vertex x equals the sum of values on edges oriented out of x, for every vertex x. The set of integral cycles is a subgroup of the integral edge space. A cycle that never takes the value zero is called nowhere zero.

Reversing the orientation of an edge negates the value of a cycle on that edge. It is in this sense that the theory is independent of the arbitrary orientation. Given any one cycle, the orientation can be chosen so that cycle takes only nonnegative values.

An integral cycle whose maximum absolute value on any edge is less than k, a positive integer, is sometimes called a k-flow on G. W.T. Tutte developed an extensive theory of nowhere-zero k-flows that is in some ways dual to that of graph coloring.

The cycle space over a field or commutative ring

The construction of the integral cycle space can be carried out for any field, abelian group, or (most generally) commutative ring (with unity) R replacing the integers. If R is a field, the cycle space is a vector space over R with dimension m - n + c, where c is the number of connected components of G. If R is any commutative ring, the cycle space is a free R-module with rank m - n + c.

When R is an abelian group such a cycle may also be called an R-flow on G. Nowhere-zero R-flows for a finite abelian group R of k elements are related to nowhere-zero integral k-flows in Tutte's theory. The number of nowhere-zero R-cycles is an evaluation of the Tutte polynomial, dual to the number of proper colorings of the graph (Tutte, 1984, Section IX.4).

References

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

• W.T. Tutte, Graph Theory, Addison-Wesley, Reading, Mass., 1984. See Chapter VIII.