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 4-cage
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 matching-generating
polynomial by
 |
(4)
|
has a true Hamilton decomposition iff 
and 
is even, and a quasi-Hamilton 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).
-Partite
Graphs into Edge-Disjoint Hamilton Circuits." Disc. Math. 14,
265-268, 1976.Saaty, T. L. and Kainen, P. C.