A complete bipartite graph, sometimes also called a complete bicolored graph (Erdős et al. 1965) or complete bigraph, is a bipartite graph (i.e., a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent) such that every pair of graph vertices in the two sets are adjacent. If there are and graph vertices in the two sets, the complete bipartite graph is denoted . The above figures show and .
is also known as the utility graph (and is the circulant graph ), and is the unique 4cage graph. is a Cayley graph. A complete bipartite graph is a circulant graph (Skiena 1990, p. 99), specifically , where is the floor function.
Special cases of are summarized in the table below.
The numbers of (directed) Hamiltonian cycles for the graph with , 2, ... are 0, 2, 12, 144, 2880, 86400, 3628800, 203212800, ... (OEIS A143248), where the th term for is given by with a factorial.
Complete bipartite graphs are graceful.
Zarankiewicz's conjecture posits a closed form for the graph crossing number of .
The independence polynomial of is given by
(1)

which has recurrence equation
(2)

the matching polynomial by
(3)

where is a Laguerre polynomial, and the matchinggenerating polynomial by
(4)

has a true Hamilton decomposition iff and is even, and a quasiHamilton decomposition iff and is odd (Laskar and Auerbach 1976; Bosák 1990, p. 124).
The complete bipartite graph illustrated above plays an important role in the novel Foucault's Pendulum by Umberto Eco (1989, p. 473; Skiena 1990, p. 143).