|
Zarankiewicz Problem
The Zarankiewicz problem, an unsolved problem in mathematics, asks for the largest possible number of edges in a bipartite graph that has a given number of vertices and has no complete bipartite subgraphs of a given size.. Reprint of 1978 Academic Press edition, . It belongs to the field of extremal graph theory, a branch of combinatorics, and is named after the Polish mathematician Kazimierz Zarankiewicz, who proposed several special cases of the problem in 1951. Problem statement A bipartite graph G=(U\cup V,E) consists of two disjoint sets of vertices U and V, and a set of edges each of which connects a vertex in U to a vertex in V. No two edges can both connect the same pair of vertices. A complete bipartite graph is a bipartite graph in which every pair of a vertex from U and a vertex from V is connected to each other. A complete bipartite graph in which U has s vertices and V has t vertices is denoted K_. If G=(U\cup V,E) is a bipartite graph, and there exists a set of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denoting ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a -cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Image:Petersen1 tiny.svg, The Petersen graph has a girth of 5 Image:Heawood_Graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4- cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure. Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Graph
Heawood is a surname. Notable people with the surname include: *Jonathan Heawood, British journalist *Percy John Heawood (1861–1955), British mathematician **Heawood conjecture **Heawood graph **Heawood number In mathematics, the Heawood number of a surface is an upper bound for the number of colors that suffice to color any graph embedded in the surface. In 1890 Heawood proved for all surfaces ''except'' the sphere that no more than : H(S)=\left\lfl ... See also * Heywood (surname) {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Double Cover
In graph theory, the bipartite double cover of an undirected graph is a bipartite, covering graph of , with twice as many vertices as . It can be constructed as the tensor product of graphs, . It is also called the Kronecker double cover, canonical double cover or simply the bipartite double of . It should not be confused with a cycle double cover of a graph, a family of cycles that includes each edge twice. Construction The bipartite double cover of has two vertices and for each vertex of . Two vertices and are connected by an edge in the double cover if and only if and are connected by an edge in . For instance, below is an illustration of a bipartite double cover of a non-bipartite graph . In the illustration, each vertex in the tensor product is shown using a color from the first term of the product () and a shape from the second term of the product (); therefore, the vertices in the double cover are shown as circles while the vertices are shown as squares. : T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, formerly titled ''Proceedings of the Cambridge Philosophical Society'', has been published since 1843. See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics education in the United Kingdom Mathematics journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Štefan Znám
Štefan Znám (9 February 1936, Veľký Blh – 17 July 1993, Bratislava) was a Slovak- Hungarian mathematician, believed to be the first to ponder Znám's problem in modern times. Znám worked in the field of number theory and 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 .... He also co-founded journal ''Matematické obzory''. External linksArticle on Štefan Znám in Matematický ústav SAV* 20th-century mathematicians Slovak mathematicians 1936 births 1993 deaths Comenius University alumni Comenius University faculty {{Europe-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pál Turán
Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. Life and education Turán was born into a Jewish family in Budapest on 18 August 1910.At the same period of time, Turán and Erdős were famous answerers in the journal '' KöMaL''. He received a teaching degree at the University of Budapest in 1933 and the PhD degree under Lipót Fejér in 1935 at Eötvös Loránd University. As a Jew, he fell victim to numerus clausus, and could not get a university job for several years. He was sent to labour service at various times from 1940-44. He is said to have been recognized and perhaps protected by a fascist guard, who, as a mathematics student, had admired Turán's work. Turán became associate professor at the University of Budapest in 1945 and ful ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vera T
Vera may refer to: Names *Vera (surname), a surname (including a list of people with the name) *Vera (given name), a given name (including a list of people and fictional characters with the name) **Vera (), archbishop of the archdiocese of Tarragona Places Spain *Vera, AlmerÃa, a municipality in the province of AlmerÃa, Andalusia *Vera de Bidasoa, a municipality in the autonomous community of Navarra *La Vera, a comarca in the province of Cáceres, Extremadura United States *Vera, Illinois, an unincorporated community * Vera, Kansas, a ghost town * Vera, Missouri, an unincorporated community * Vera, Oklahoma, a town *Vera, Texas, an unincorporated community * Vera, Virginia, an unincorporated community *Veradale, Washington, originally known as Vera, CDP Elsewhere *Vera, Santa Fe, a city in the province of Santa Fe, Argentina *Vera Department, an administrative subdivision (departamento) of the province of Santa Fe * Vera, Mato Grosso, Brazil, a municipality *Cape Vera, Nun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element of that is less than or equal to every element of . A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below) are also used in the mathematical literature for sets that have upper (respectively lower) bounds. Examples For example, is a lower bound for the set (as a subset of the integers or of the real numbers, etc.), and so is . On the other hand, is not a lower bound for since it is not smaller than every element in . The set has as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that . Every subset of the natural numbers has a lowe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Degree (graph Theory)")