TOPICS

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.

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).

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

Explore with Wolfram|Alpha

More things to try:

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."

Referenced on Wolfram|Alpha

Complete Bipartite Graph

Cite this as:

Weisstein, Eric W. "Complete Bipartite Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompleteBipartiteGraph.html