Complete bipartite graph

In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.<ref name="bm">Template:Citation.</ref><ref name="d">Template:Citation. Electronic edition, page 17.</ref>

Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher.<ref name="knuth"/><ref>Template:Citation.</ref> Llull himself had made similar drawings of complete graphs three centuries earlier.<ref name="knuth">Template:Citation. </ref>

Definition

A complete bipartite graph is a graph whose vertices can be partitioned into two subsets Template:Math and Template:Math 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 bipartite graph Template:Math such that for every two vertices Template:Math andTemplate:Math, Template:Math is an edge in Template:Mvar. A complete bipartite graph with partitions of size Template:Math and Template:Math, is denoted Template:Math;<ref name="bm"/><ref name="d"/> every two graphs with the same notation are isomorphic.

Examples

File:Star graphs.svg

The star graphs Template:Math, Template:Math, Template:Math, and Template:Math.

File:Zarankiewicz K4 7.svg

A complete bipartite graph of Template:Math showing that Turán's brick factory problem with 4 storage sites (yellow spots) and 7 kilns (blue spots) requires 18 crossings (red dots)

• For any Template:Mvar, Template:Math is called a star.<ref name="d"/> All complete bipartite graphs which are trees are stars.
  • The graph Template:Math is called a claw, and is used to define the claw-free graphs.<ref>Template:Citation. Corrected reprint of the 1986 original.</ref>
• The graph Template:Math 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 Template:Math.<ref>Template:Citation.</ref>
• 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

Example Template:Math complete bipartite graphs<ref>Coxeter, Regular Complex Polytopes, second edition, p.114</ref>

Template:Math	Template:Math	Template:Math
File:Complex polygon 2-4-3-bipartite graph.png	File:Complex polygon 2-4-4 bipartite graph.png	File:Complex polygon 2-4-5-bipartite graph.png
File:Complex polygon 2-4-3.png 3 edge-colorings	File:Complex polygon 2-4-4.png 4 edge-colorings	File:Complex polygon 2-4-5.png 5 edge-colorings
Regular complex polygons of the form Template:Math have complete bipartite graphs with Template:Math vertices (red and blue) and Template:Math 2-edges. They also can also be drawn as Template:Mvar edge-colorings.

• Given a bipartite graph, testing whether it contains a complete bipartite subgraph Template:Math for a parameter Template:Mvar is an NP-complete problem.<ref>Template:Citation.</ref>
• A planar graph cannot contain Template:Math as a minor; an outerplanar graph cannot contain Template:Math as a minor (These are not sufficient conditions for planarity and outerplanarity, but necessary). Conversely, every nonplanar graph contains either Template:Math or the complete graph Template:Math as a minor; this is Wagner's theorem.<ref>Template:Harvnb</ref>
• Every complete bipartite graph. Template:Math is a Moore graph and a Template:Math-cage.<ref>Template:Citation.</ref>
• The complete bipartite graphs Template:Math and Template:Math 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 Template:Mvar-partite graphs and graphs that avoid larger cliques as subgraphs in Turán's theorem, and these two complete bipartite graphs are examples of Turán graphs, the extremal graphs for this more general problem.<ref>Template:Citation.</ref>
• The complete bipartite graph Template:Math has a vertex covering number of Template:Math and an edge covering number of Template:Math
• The complete bipartite graph Template:Math has a maximum independent set of size Template:Math
• The adjacency matrix of a complete bipartite graph Template:Math has eigenvalues Template:Math, Template:Math and 0; with multiplicity 1, 1 and Template:Math respectively.<ref>Template:Harvtxt, p. 266.</ref>
• The Laplacian matrix of a complete bipartite graph Template:Math has eigenvalues Template:Math, Template:Mvar, Template:Mvar, and 0; with multiplicity 1, Template:Math, Template:Math and 1 respectively.
• A complete bipartite graph Template:Math has Template:Math spanning trees.<ref>Template:Citation.</ref>
• A complete bipartite graph Template:Math has a maximum matching of size Template:Math
• A complete bipartite graph Template:Math has a proper [[edge coloring|Template:Mvar-edge-coloring]] corresponding to a Latin square.<ref>Template:Citation.</ref>
• 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.<ref>Template:Citation.</ref>

See also

• Biclique-free graph, a class of sparse graphs defined by avoidance of complete bipartite subgraphs
• Crown graph, a graph formed by removing a perfect matching from a complete bipartite graph
• Complete multipartite graph, a generalization of complete bipartite graphs to more than two sets of vertices
• Biclique attack