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."