An ( N, M, D, K, e )- disperser is a bipartite graph with N vertices on the left side, each with degree D, and M vertices on the right side, such that every subset of K vertices on the left side is connected to more than ( 1 − e ) M vertices on the right.

An-extractor is a bipartite graph with nodes on the left and nodes on the right such that each node on the left has neighbors ( on the right ), which has the added property that

* A graph is planar if it contains as a minor neither the complete bipartite graph ( See the Three-cottage problem ) nor the complete graph.

* How many perfect matchings are there for a given bipartite graph?

It was known before that the decision problem " Is there a perfect matching for a given bipartite graph?

The corresponding question " How many perfect matchings does the given bipartite graph have?

* Every tree is a bipartite graph and a median graph.

The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side ; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.

* Barnette's conjecture, a still-open refinement of Tait's conjecture stating that every bipartite cubic polyhedral graph is Hamiltonian.

In more formal graph-theoretic terms, the problem asks whether the complete bipartite graph K < sub > 3, 3 </ sub > is planar.

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

Pál Turán's " brick factory problem " asks more generally for a formula for the minimum number of crossings in a drawing of the complete bipartite graph K < sub > a, b </ sub > in terms of the numbers of vertices a and b on the two sides of the bipartition.

The utility graph K < sub > 3, 3 </ sub > is, like all other complete bipartite graphs, a well-covered graph, meaning that every maximal independent set has the same size.

If at any step a vertex has ( visited ) neighbours with the same label as itself, then the graph is not bipartite.

If the search ends without such a situation occurring, then the graph is bipartite.

It consists of finding a maximum weight matching in a weighted bipartite graph.

Displayed are parts of the ( disjoint ) sets A and B together with parts of the mappings f and g. If the set A ∪ B, together with the two maps, is interpreted as a directed graph, then this bipartite graph has several connected components.

* Bipartite graph, a graph is bipartite if it has no odd cycles

In the simplest case, one may consider an Ising model on an bipartite lattice, e. g. the simple cubic lattice, with couplings between spins at nearest neighbor sites.

The elastic, delicate radular membrane may be a single tongue, or may split into two ( bipartite ).

Based on the bipartite nature of the radular dentition pattern in solenogasters, larval gastropods and larval polyplacophora, it has been postulated that the ancestral mollusc bore a bipartite radula ( although the radular membrane may not have been bipartite ).

A Petri net is a directed bipartite graph, in which the nodes represent transitions ( i. e. events that may occur, signified by bars ) and places ( i. e. conditions, signified by circles ).

A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and ( x < sub > 1 </ sub >, e < sub > 1 </ sub >) are connected with an edge if and only if vertex x < sub > 1 </ sub > is contained in edge e < sub > 1 </ sub > in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above.

Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer bipartite, but is rather just some general directed graph.

If a bipartite graph is not connected, it may have more than one bipartition ; in this case, the notation is helpful in specifying one particular bipartition that may be of importance in an application.

Biadjacency matrices may be used to describe equivalences between bipartite graphs, hypergraphs, and directed graphs.

As a special case of this correspondence between bipartite graphs and hypergraphs, any multigraph ( a graph in which there may be two or more edges between the same two vertices ) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have degree two.

A similar reinterpretation of adjacency matrices may be used to show a one-to-one correspondence between directed graphs ( on a given number of labeled vertices, allowing self-loops ) and balanced bipartite graphs, with the same number of vertices on both sides of the bipartition.

For the intersection graphs of line segments or other simple shapes in the Euclidean plane, it is possible to test whether the graph is bipartite and return either a two-coloring or an odd cycle in time, even though the graph itself may have as many as edges.

The concept of the line graph of G may naturally be extended to the case where G is a multigraph, although in that case Whitney's uniqueness theorem no longer holds ; for instance a complete bipartite graph K < sub > 1, n </ sub > has the same line graph as the dipole graph and Shannon multigraph with the same number of edges.

For some graphs, such as bipartite graphs and high-degree planar graphs, the number of colors is always, and for multigraphs, the number of colors may be as large as.

When a graph has a 2-join, it may be decomposed into induced subgraphs called " blocks ", by replacing one of the two subsets of vertices by a shortest path within that subset that connects one of the two complete bipartite graphs to the other ; when no such path exists, the block is formed instead by replacing one of the two subsets of vertices by two vertices, one for each complete bipartite subgraph.

A Chelsky sequence may, therefore, be part of the downstream basic cluster of a bipartite NLS.

It was also foreseen that there would be clauses inserted in bipartite international treaties which would allow the referral of disputes to the ECJ ; this indeed occurred, with such provisions found in treaties between Czechoslovakia and Austria, and between Czechoslovakia and Poland.

While it is true that a bipartite quantum state must be entangled in order for it to produce non-local correlations, there exist entangled states that do not produce such correlations.

The general bipartite case has been shown to be NP-hard.

Traditional Islam views the world as bipartite, consisting of the House of Islam, that is, where people live under the Islamic law-the Shariah-and the House of War, that is, where the people do not live under Islamic law, which must be proselytized ( see da ' wah ) using whatever resources available, including, in some traditionalist and conservative interpretations, the use of violence, as holy struggle in the path of Allah, to either convert its inhabitants to Islam, or to rule them under the Shariah ( cf.

Using this method, he showed how to solve the Hamiltonian cycle problem in arbitrary n-vertex graphs by a Monte Carlo algorithm in time O ( 1. 657 < sup > n </ sup >); for bipartite graphs this algorithm can be further improved to time O ( 1. 414 < sup > n </ sup >).

However, putting all of these conditions together, it remains open whether 3-connected 3-regular bipartite planar graphs must always contain a Hamiltonian cycle, in which case the problem restricted to those graphs could not be NP-complete ; see Barnette's conjecture.

A square matrix can also be viewed as the biadjacency matrix of a bipartite graph which has vertices on one side and on the other side, with representing the weight of the edge from vertex to vertex.

In particular, there is a bipartite " incidence graph " or " Levi graph " corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.

A hypergraph is bipartite if and only if its vertices can be partitioned into two classes U and V in such a way that each hyperedge contains at least one vertex from both classes.

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

This problem can be modeled as a dominating set problem in a bipartite graph that has a vertex for each train and each station and an edge for