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
is a Cayley graph. A complete bipartite graph
is a circulant graph (Skiena 1990, p. 99),
is the floor function.
Special cases of
are summarized in the table below.
The numbers of (directed)
Hamiltonian cycles for the graph
2, ... are 0, 2, 12, 144, 2880, 86400, 3628800, 203212800, ... (OEIS A143248),
where the th
is given by
Complete bipartite graphs are
Zarankiewicz's conjecture posits a closed form for the graph crossing number of .
independence polynomial of is given by
which has recurrence equation
matching polynomial by
is a Laguerre polynomial, and the matching-generating
has a true Hamilton decomposition iff
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
by Umberto Eco (1989, p. 473; Skiena 1990, p. 143).
See also Bipartite Graph
Cocktail Party Graph
Complete k-Partite Graph
Complete Tripartite Graph
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References Bosák, J. New York: Springer, 1990. Decompositions of Graphs. Chia, G. L. and Sim,
K. A. "On the Skewness of the Join of Graphs." Disc. Appl. Math. 161,
2405-2409, 2013. Eco, U. San Diego: Harcourt Brace Jovanovich, p. 473, 1989. Foucault's
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
Graphs into Edge-Disjoint Hamilton Circuits." Disc. Math. 14,
265-268, 1976. Saaty, T. L. and Kainen, P. C. New York: Dover, p. 12, 1986. The
Four-Color Problem: Assaults and Conquest. Skiena,
MA: Addison-Wesley, 1990. Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Sloane, N. J. A. Sequence A143248
in "The On-Line Encyclopedia of Integer Sequences." Referenced
on Wolfram|Alpha Complete Bipartite Graph
Cite this as:
Weisstein, Eric W. "Complete Bipartite Graph."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/CompleteBipartiteGraph.html