# Bipartite graph

In the mathematical field of graph theory, a bipartite graph is a special graph where the set of vertices can be divided into two disjoint sets with two vertices of the same set never sharing an edge.

Bipartite graphs are useful for modelling matching problems. An example is a job matching problem. Suppose we have a set of people P and a set of jobs J, with not all people suitable for all jobs. We can model this as a graph with P + J the set of vertices. If a person [itex]p_i[itex] is suitable for a certain job [itex]j_i[itex] there is a edge between [itex]p_i[itex] and [itex]j_i[itex] in the graph.

The marriage theorem provides a characterization of bipartite graphs which allow perfect matchings.

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## Definitions

A simple undirected graph [itex]G:=(V,E)[itex] is called bipartite if there exists a partition of the vertex set [itex]V=V_1 \cup V_2 [itex] so that both [itex]V_1[itex] and [itex]V_2[itex] are independent sets. We often write [itex]G:=(V_1 + V_2, E)[itex] to denote a bipartite graph with partitions [itex]V_1[itex] and [itex]V_2[itex].

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