Bipartite graph

Example of a bipartite graph without cycles

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets ${\displaystyle U}$ and ${\displaystyle V}$, that is, every edge connects a vertex in ${\displaystyle U}$ to one in ${\displaystyle V}$. Vertex sets ${\displaystyle U}$ and ${\displaystyle V}$ are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.^[1][2]

The two sets ${\displaystyle U}$ and ${\displaystyle V}$ may be thought of as a coloring of the graph with two colors: if one colors all nodes in ${\displaystyle U}$ blue, and all nodes in ${\displaystyle V}$ red, each edge has endpoints of differing colors, as is required in the graph coloring problem.^[3][4] 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 red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color.

One often writes ${\displaystyle G=(U,V,E)}$ to denote a bipartite graph whose partition has the parts ${\displaystyle U}$ and ${\displaystyle V}$, with ${\displaystyle E}$ denoting the edges of the graph. If a bipartite graph is not connected, it may have more than one bipartition;^[5] in this case, the ${\displaystyle (U,V,E)}$ notation is helpful in specifying one particular bipartition that may be of importance in an application. If ${\displaystyle |U|=|V|}$, that is, if the two subsets have equal cardinality, then ${\displaystyle G}$ is called a balanced bipartite graph.^[3] If all vertices on the same side of the bipartition have the same degree, then ${\displaystyle G}$ is called biregular.