Графтардың жұп дәрежелі кішіграфтары

All even degree subgraphs of a graph
a concept in graph theory
In graph theory, a branch of mathematics, the (binary) cycle space of an undirected graph is the set of its even degree 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.

The cycle space of a graph can be described with increasing levels of mathematical sophistication as a set of subgraphs, as a binary vector space, or as a homology group.

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 has the elements of S as its 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, for the purposes of defining cycle spaces, 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 in this sense. The cycle space of a graph is the collection of its Eulerian spanning subgraphs. The cycle space, also, has an algebraic structure, but a more restrictive one. The union or intersection of two Eulerian subgraphs may fail to be Eulerian. However, the symmetric difference of two Eulerian subgraphs
(the graph consisting of the edges that belong to exactly one of the two given graphs) is again Eulerian. This field has two elements, 0 and 1, and its addition and multiplication operations can be described as the familiar addition and multiplication of integers, taken modulo 2. A vector space consists of a set of elements together with an addition and scalar multiplication operation satisfying certain properties generalizing the properties of the familiar real vector spaces. For the cycle space, the elements of the vector space are the Eulerian subgraphs, the addition operation is symmetric differencing, multiplication by the scalar 1 is the identity operation, and multiplication by the scalar 0 takes every element to the empty graph, which forms the additive identity element for the cycle space. The edge space is also a vector space over with the symmetric difference as addition. As vector spaces, the cycle space and the cut space of the graph (the family of edge sets that span the cuts of the graph) are the orthogonal complements of each other within the edge space. This means that a set of edges in a graph forms a cut if and only if every Eulerian subgraph has an even number of edges in common with , and forms an Eulerian subgraph if and only if every cut has an even number of edges in common with .

As a vector space, the dimension of the cycle space of a graph with vertices, edges, and connected components is This number can be interpreted topologically as the first Betti number of the graph. every Eulerian subgraph of a given graph can be decomposed into simple cycles, subgraphs in which all vertices have degree zero or two and in which the degree two vertices form a connected set. Therefore, it is always possible to find a basis in which the basis elements are themselves all simple cycles. Such a basis is called a cycle basis of the given graph. More strongly, it is always possible to find a basis in which the basis elements are induced cycles or even (in a 3 vertex connected graph) induced cycles whose removal does not separate the remaining graph.

One way of constructing a cycle basis is to form a maximal forest of the graph, and then for each edge that does not belong to the forest, form a cycle consisting of together with the path in the forest connecting the endpoints of The cycles formed in this way are linearly independent (each one contains an edge that does not belong to any of the other cycles) and has the correct size to be a basis, so it necessarily is a basis. A basis formed in this way is called a fundamental cycle basis (with respect to the chosen forest).

If the edges of a graph are given real number weights, the weight of a subgraph may be computed as the sum of the weights of its edges. The minimum weight basis of the cycle space is necessarily a cycle basis, and can be constructed in polynomial time. It follows from this that the set of bounded faces of the embedding forms a cycle basis for the planar graph: removing the unbounded face from this set of cycles reduces the number of ways each Eulerian subgraph can be generated from two to exactly one.

Mac Lane's planarity criterion, named after Saunders Mac Lane, characterizes planar graphs in terms of their cycle spaces and cycle bases. It states that a finite undirected graph is planar if and only if the graph has a cycle basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, a cycle basis formed by the set of bounded faces of an embedding necessarily has this property: each edge participates only in the basis cycles for the two faces it separates. Conversely, if a cycle basis has at most two cycles per edge, then its cycles can be used as the set of bounded faces of a planar embedding of its graph.

The cycle space of a planar graph is the cut space of its dual graph, and vice versa. The minimum weight cycle basis for a planar graph is not necessarily the same as the basis formed by its bounded faces: it can include cycles that are not faces, and some faces may not be included as cycles in the minimum weight cycle basis. There exists a minimum weight cycle basis in which no two cycles cross each other: for every two cycles in the basis, either the cycles enclose disjoint subsets of the bounded faces, or one of the two cycles encloses the other one. Following the duality between cycle spaces and cut spaces, this basis for a planar graph corresponds to a Gomory–Hu tree of the dual graph, a minimum weight basis for its cut space.

In planar graphs, colorings with distinct colors are dual to nowhere zero flows over the ring of integers modulo In this duality, the difference between the colors of two adjacent regions is represented by a flow value across the edge separating the regions. In particular, the existence of nowhere zero 4 flows is equivalent to the four color theorem. The snark theorem generalizes this result to nonplanar graphs.