# Graph, bipartite

*bichromatic graph*

A graph whose set of vertices can be partitioned into two disjoint sets and (i.e. , ) so that each edge connects some vertex of with some vertex of . A graph is bipartite if and only if all its simple cycles have even length. Another frequently used definition of a bipartite graph is a graph in which two subsets and of vertices (parts) are given in advance. Bipartite graphs are convenient for the representation of binary relations between elements of two different types — e.g. the elements of a given set and a subset of it yield the relation of "membership of an element to a subset" , for executors and types of jobs one has the relation "a given executor can carry out a given job" , etc.

An important problem concerning bipartite graphs is the study of matchings, that is, families of pairwise non-adjacent edges. Such problems occur, for example, in the theory of scheduling (partitioning of the edges of a bipartite graph into a minimal number of disjoint matchings), in the problem of assignment (finding the maximum number of elements in a matching), etc. The cardinality of the maximum matching in a bipartite graph is

where is the number of vertices of adjacent to at least one vertex of . A complete bipartite graph is a bipartite graph in which any two vertices belonging to different subsets are connected by an edge (e.g. the graph , see Graph, planar, Figure 1).

A generalization of the concept of a bipartite graph is the concept of a -partite graph, i.e. a graph in which the vertices are partitioned into subsets so that each edge connects vertices belonging to different subsets.

#### References

[1] | O. Ore, "Theory of graphs" , Amer. Math. Soc. (1962) |

#### Comments

#### References

[a1] | F. Harary, "Graph theory" , Addison-Wesley (1969) pp. Chapt. 9 |

[a2] | R.J. Wilson, "Introduction to graph theory" , Longman (1985) |

**How to Cite This Entry:**

Graph, bipartite.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Graph,_bipartite&oldid=16372