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 and (that is, and are each independent sets) such that every edge connects a vertex in to one in . Vertex set and are often denoted as partite sets. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.

The two sets and may be thought of as a coloring of the graph with two colors: if one colors all nodes in blue, and all nodes in 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 non-bipartite 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 to denote a bipartite graph whose partition has the parts and , with denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition; in this case, the notation is helpful in specifying one particular bipartition that may be of importance in an application. If , that is, if the two subsets have equal cardinality, then is called a balanced bipartite graph. If all vertices on the same side of the bipartition have the same degree, then is called biregular.