Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

The two sets U and V may be thought of as a coloring of the graph with two colors: if we color all nodes in U blue, and all nodes in V green, each edge has endpoints of differing colors, as is required in the graph coloring problem. In contrast, such a coloring is impossible in the case of a nonbipartite graph, such as a triangle: after one node is colored blue and another green, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color.

One often writes G = (U, V, E) to denote a bipartite graph whose partition has the parts U and V. If |U| =|V|, that is, if the two subsets have equal cardinality, then G is called a balanced bipartite graph.

Examples

Any graph with no odd cycles is bipartite. As a consequence of this:
- Every tree is bipartite.
- Cycle graphs with an even number of vertices are bipartite.
- Any planar graph where all the faces in its planar representation consist of an even number of edges is bipartite.

Testing bipartiteness

If a bipartite graph is connected, its bipartition can be defined by the parity of the distances from any arbitrarily chosen vertex v: one subset consists of the vertices at even distance to v and the other subset consists of the vertices at odd distance to v.

Thus, one may efficiently test whether a graph is bipartite by using this parity technique to assign vertices to the two subsets U and V, separately within each connected component of the graph, and then examine each edge to verify that it has endpoints assigned to different subsets.

Applications

Bipartite graphs are useful for modelling matching problems. An example of bipartite graph is a job matching problem. Suppose we have a set P of people and a set J of jobs, with not all people suitable for all jobs. We can model this as a bipartite graph (P, J, E). If a person p_x is suitable for a certain job j_y there is an edge between p_x and j_y in the graph. The marriage theorem provides a characterization of bipartite graphs which allow perfect matchings.

Bipartite graphs are extensively used in modern Coding theory, especially to decode codewords received from the channel. Factor graphs and Tanner graphs are examples of this.

In computer science, a Petri net is a mathematical modelling tool used in analysis and simulations of concurrent systems. A system is modelled as a bipartite directed graph with two sets of nodes: A set of "place" nodes that contain resources, and a set of "event" nodes which generate and/or consume resources. There are additional constraints on the nodes and edges that constrain the behavior of the system. Petri nets utilize the properties of bipartite directed graphs and other properties to allow mathematical proofs of the behavior of systems while also allowing easy implementation of simulations of the system.

In projective geometry, Levi graphs are a form of bipartite graph used to model the incidences between points and lines in a configuration.

Modelling multigraphs and hypergraphs

Bipartite graphs can model entirely the more general multigraph. Given a multigraph M, take U as the vertex set of M and take V as the edge set of M. Then join an element of V to precisely the two elements of U which are the ends of that edge in M. Thus every multigraph is described entirely by a bipartite graph which is one-sided regular of degree 2, and vice versa.

Similarly, every directed hypergraph can be represented as a bipartite digraph. Take U as the vertex set in the hypergraph, and V as set of edges. For each $u\in U$ and $v \in V$, connect u to v if the hypergraph edge v contains u as an input, and connect v to u if v contains u as an output.

Properties

A graph is bipartite if and only if it does not contain an odd cycle. Therefore, a bipartite graph cannot contain a clique of size 3 or more.
A graph is bipartite if and only if it is 2-colorable, (i.e. its chromatic number is less than or equal to 2).
The size of minimum vertex cover is equal to the size of the maximum matching (König's theorem).
The size of the maximum independent set plus the size of the maximum matching is equal to the number of vertices.
For a connected bipartite graph the size of the minimum edge cover is equal to the size of the maximum independent set.
For a connected bipartite graph the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices.
Every bipartite graph is a perfect graph.
The spectrum of a graph is symmetric if and only if it's a bipartite graph.

External links

Information System on Graph Class Inclusions: bipartite graph

Weisstein, Eric W. "Bipartite Graph". MathWorld.