In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...

, a complete bipartite graph or biclique is a special kind of

bipartite graph In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V a ...

where every

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

of the first set is connected to every vertex of the second set..
Electronic edition
page 17. Graph theory itself is typically dated as beginning with

Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...

's 1736 work on the Seven Bridges of Königsberg. However,

drawing Drawing is a form of visual art in which an artist uses instruments to mark paper or other two-dimensional surface. Drawing instruments include graphite pencils, pen and ink, various kinds of paints, inked brushes, colored pencils, crayo ...

s of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of

Ramon Llull Ramon Llull (; c. 1232 – c. 1315/16) was a philosopher, theologian, poet, missionary, and Christian apologist from the Kingdom of Majorca. He invented a philosophical system known as the ''Art'', conceived as a type of universal logic to pro ...

edited by

Athanasius Kircher Athanasius Kircher (2 May 1602 – 27 November 1680) was a German Jesuit scholar and polymath who published around 40 major works, most notably in the fields of comparative religion, geology, and medicine. Kircher has been compared to fe ...

. Llull himself had made similar drawings of complete graphs three centuries earlier..

Definition

A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a

such that for every two vertices and, is an edge in . A complete bipartite graph with partitions of size and , is denoted ; every two graphs with the same notation are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

Examples

* For any , is called a

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

. All complete bipartite graphs which are

trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...

are stars. ** The graph is called a

claw A claw is a curved, pointed appendage found at the end of a toe or finger in most amniotes (mammals, reptiles, birds). Some invertebrates such as beetles and spiders have somewhat similar fine, hooked structures at the end of the leg or tarsus ...

, and is used to define the

claw-free graph In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph. A claw is another name for the complete bipartite graph ''K''1,3 (that is, a star graph comprising three edges, three leaves, ...

s. * The graph is called the utility graph. This usage comes from a standard mathematical puzzle in which three utilities must each be connected to three buildings; it is impossible to solve without crossings due to the nonplanarity of . * The maximal bicliques found as subgraphs of the digraph of a relation are called concepts. When a lattice is formed by taking meets and joins of these subgraphs, the relation has an Induced concept lattice. This type of analysis of relations is called formal concept analysis.

Properties

*Given a bipartite graph, testing whether it contains a complete bipartite subgraph for a parameter is an NP-complete problem. *A

planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...

cannot contain as a minor; an outerplanar graph cannot contain as a minor (These are not

sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

s for planarity and outerplanarity, but necessary). Conversely, every nonplanar graph contains either or the complete graph as a minor; this is

Wagner's theorem In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither ''K''5 (the complete graph on five ...

. *Every complete bipartite graph. is a

Moore graph In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is and its diameter is , its girth must ...

and a - cage. *The complete bipartite graphs and have the maximum possible number of edges among all triangle-free graphs with the same number of vertices; this is Mantel's theorem. Mantel's result was generalized to -partite graphs and graphs that avoid larger cliques as subgraphs in Turán's theorem, and these two complete bipartite graphs are examples of

Turán graph The Turán graph, denoted by T(n,r), is a complete multipartite graph; it is formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to di ...

s, the extremal graphs for this more general problem. *The complete bipartite graph has a vertex covering number of and an edge covering number of *The complete bipartite graph has a maximum independent set of size *The

adjacency matrix In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simp ...

of a complete bipartite graph has eigenvalues , and 0; with multiplicity 1, 1 and respectively. *The

Laplacian matrix In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph La ...

of a complete bipartite graph has eigenvalues , , , and 0; with multiplicity 1, , and 1 respectively. *A complete bipartite graph has

spanning tree In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is ...

s. *A complete bipartite graph has a maximum matching of size *A complete bipartite graph has a proper -edge-coloring corresponding to a Latin square. *Every complete bipartite graph is a modular graph: every triple of vertices has a median that belongs to shortest paths between each pair of vertices..

References

{{reflist, 30em Parametric families of graphs

Definition

Examples

Properties

See also

References