Complete Bipartite Graph
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.
 |
path graph  |
 |
path graph  |
 |
claw graph |
 |
star graph  |
 |
square graph  |
 |
utility graph |
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).
REFERENCES
Bosák, J. Decompositions of Graphs. New York: Springer, 1990.
Chia, G. L. and Sim, K. A. "On the Skewness of the Join of Graphs." Disc. Appl. Math. 161, 2405-2409, 2013.
Eco, U. Foucault's Pendulum. San Diego: Harcourt Brace Jovanovich, p. 473, 1989.
Erdős, P.; Harary, F.; and Tutte, W. T. "On the Dimension of a Graph." Mathematika 12, 118-122, 1965.
Laskar, R. and Auerbach, B. "On Decomposition of 
-Partite Graphs into Edge-Disjoint Hamilton Circuits." Disc. Math. 14, 265-268, 1976.
Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 12, 1986.
Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Sloane, N. J. A. Sequence A143248 in "The On-Line Encyclopedia of Integer Sequences."
