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 p and q graph vertices in the two sets, the complete bipartite graph is denoted K_(p,q). The above figures show K_(3,2) and K_(2,5).


K_(3,3) is also known as the utility graph (and is the circulant graph Ci_(1,3)(6)), and is the unique 4-cage graph. K_(4,4) is a Cayley graph. A complete bipartite graph K_(n,n) is a circulant graph (Skiena 1990, p. 99), specifically Ci_(1,3,...,2|_n/2_|+1)(n), where |_x_| is the floor function.

Special cases of K_(m,n) are summarized in the table below.

The numbers of (directed) Hamiltonian cycles for the graph K_(n,n) with n=1, 2, ... are 0, 2, 12, 144, 2880, 86400, 3628800, 203212800, ... (OEIS A143248), where the nth term for n>1 is given by n!(n-1)! with n! a factorial.

Complete bipartite graphs are graceful.

Zarankiewicz's conjecture posits a closed form for the graph crossing number of K_(m,n).

The independence polynomial of K_(n,n) is given by


which has recurrence equation


the matching polynomial by


where L_n(x) is a Laguerre polynomial, and the matching-generating polynomial by


K_(m,n) has a true Hamilton decomposition iff m=n and m is even, and a quasi-Hamilton decomposition iff m=n and m is odd (Laskar and Auerbach 1976; Bosák 1990, p. 124).


The complete bipartite graph K_(18,18) illustrated above plays an important role in the novel Foucault's Pendulum by Umberto Eco (1989, p. 473; Skiena 1990, p. 143).

See also

Bipartite Graph, Cage Graph, Cocktail Party Graph, Complete Graph, Complete k-Partite Graph, Complete Tripartite Graph, Crown Graph, k-Partite Graph, Thomassen Graphs, Utility Graph, Zarankiewicz's Conjecture

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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 r-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."

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Complete Bipartite Graph

Cite this as:

Weisstein, Eric W. "Complete Bipartite Graph." From MathWorld--A Wolfram Web Resource.

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