|
Forbidden Subgraph Problem
In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph G, find the maximal number of edges \operatorname(n,G) an n-vertex graph can have such that it does not have a Glossary of graph theory#Subgraphs, subgraph graph isomorphism, isomorphic to G. In this context, G is called a forbidden subgraph.''Combinatorics: Set Systems, Hypergraphs, Families of Vectors and Probabilistic Combinatorics'', Béla Bollobás, 1986, p. 53, 54/ref> An equivalent problem is how many edges in an n-vertex graph guarantee that it has a subgraph isomorphic to G? Definitions The extremal number \operatorname(n,G) is the maximum number of edges in an n-vertex graph containing no subgraph isomorphic to G. K_r is the complete graph on r vertices. T(n,r) is the Turán graph: a complete Multipartite graph, r-partite graph on n vertices, with vertices distributed between parts as equally as possible. The chromatic number \chi(G) of G is the minimum number of colors neede ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Graph Theory
Extremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative connections between various graph properties, both global (such as the number of vertices and edges) and local (such as the existence of specific subgraphs), and problems in extremal graph theory can often be formulated as optimization problems: how big or small a parameter of a graph can be, given some constraints that the graph has to satisfy? A graph that is an optimal solution to such an optimization problem is called an extremal graph, and extremal graphs are important objects of study in extremal graph theory. Extremal graph theory is closely related to fields such as Ramsey theory, spectral graph theory, computational complexity theory, and additive combin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noga Alon
Noga Alon ( he, נוגה אלון; born 17 February 1956) is an Israeli mathematician and a professor of mathematics at Princeton University noted for his contributions to combinatorics and theoretical computer science, having authored hundreds of papers. Academic background Alon is a Professor of Mathematics at Princeton University and a Baumritter Professor Emeritus of Mathematics and Computer Science at Tel Aviv University, Israel. He graduated from the Hebrew Reali School in 1974 and received his Ph.D. in Mathematics at the Hebrew University of Jerusalem in 1983 and had visiting positions in various research institutes including MIT, The Institute for Advanced Study in Princeton, IBM Almaden Research Center, Bell Labs, Bellcore and Microsoft Research. He serves on the editorial boards of more than a dozen international journals; since 2008 he is the editor-in-chief of ''Random Structures and Algorithms''. He has given lectures in many conferences, including plenary addresses ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgraph Isomorphism Problem
In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs ''G'' and ''H'' are given as input, and one must determine whether ''G'' contains a subgraph that is isomorphic to ''H''. Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a Hamiltonian cycle, and is therefore NP-complete. However certain other cases of subgraph isomorphism may be solved in polynomial time. Sometimes the name subgraph matching is also used for the same problem. This name puts emphasis on finding such a subgraph as opposed to the bare decision problem. Decision problem and computational complexity To prove subgraph isomorphism is NP-complete, it must be formulated as a decision problem. The input to the decision problem is a pair of graphs ''G'' and ''H''. The answer to the problem is positive if ''H'' is isomorphic to a subgraph of ''G'', and negative otherwise. Formal question ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán Number
In mathematics, the Turán number T(''n'',''k'',''r'') for ''r''-uniform hypergraphs of order ''n'' is the smallest number of ''r''-edges such that every induced subgraph on ''k'' vertices contains an edge. This number was determined for ''r'' = 2 by , and the problem for general ''r'' was introduced in . The paper gives a survey of Turán numbers. Definitions Fix a set ''X'' of ''n'' vertices. For given ''r'', an ''r''-edge or block is a set of ''r'' vertices. A set of blocks is called a Turán (''n'',''k'',''r'') system (''n'' ≥ ''k'' ≥ ''r'') if every ''k''-element subset of ''X'' contains a block. The Turán number T(''n'',''k'',''r'') is the minimum size of such a system. Example The complements of the lines of the Fano plane form a Turán (7,5,4)-system. T(7,5,4) = 7. Relations to other combinatorial designs It can be shown that ::T(n,k,r) \geq \binom ^. Equality holds if and only if there exists a Steiner system S(''n'' - ''k'', ''n'' - ''r'', ''n''). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Hajnal Conjecture
In graph theory, a branch of mathematics, the Erdős–Hajnal conjecture states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is named for Paul Erdős and András Hajnal. More precisely, for an arbitrary undirected graph H, let \mathcal_H be the family of graphs that do not have H as an induced subgraph. Then, according to the conjecture, there exists a constant \delta_H > 0 such that the n-vertex graphs in \mathcal_H have either a clique or an independent set of size \Omega(n^). An equivalent statement to the original conjecture is that, for every graph H, the H-free graphs all contain polynomially large perfect induced subgraphs. Graphs without large cliques or independent sets In contrast, for random graphs in the Erdős–Rényi model with edge probability 1/2, both the maximum clique and the maximum independent set are much smaller: their size is proportional to the logarithm of n, rather than growin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biclique-free Graph
In graph theory, a branch of mathematics, a -biclique-free graph is a graph that has no -vertex complete bipartite graph as a subgraph. A family of graphs is biclique-free if there exists a number such that the graphs in the family are all -biclique-free. The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and have also been used in parameterized complexity. Properties Sparsity According to the Kővári–Sós–Turán theorem, every -vertex -biclique-free graph has edges, significantly fewer than a dense graph would have. Conversely, if a graph family is defined by forbidden subgraphs or closed under the operation of taking subgraphs, and does not include dense graphs of arbitrarily large size, it must be -biclique-free for some , for otherwise it would include large dense complete bipartite graphs. As a lower bound, conjectured that every maximal -biclique-free bipartite graph ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, an undirected hypergraph H is a pair H = (X,E) where X is a set of elements called ''nodes'' or ''vertices'', and E is a set of non-empty subsets of X called '' hyperedges'' or ''edges''. Therefore, E is a subset of \mathcal(X) \setminus\, where \mathcal(X) is the power set of X. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''. A directed hypergraph differs in that its hyperedges are not sets, but ordered pairs of subsets of X, with each pair's first and second entries constituting the tail and head of the hyperedge respectively. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pigeonhole Principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there must be at least two right-handed gloves, or at least two left-handed gloves, because there are three objects, but only two categories of handedness to put them into. This seemingly obvious statement, a type of counting argument, can be used to demonstrate possibly unexpected results. For example, given that the population of London is greater than the maximum number of hairs that can be present on a human's head, then the pigeonhole principle requires that there must be at least two people in London who have the same number of hairs on their heads. Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canad
{{Disambiguation, geo ...
Canad may refer to: * Sanjak of Çanad, an Ottoman-era district * Magyarcsanád, known in Serbian as Čanad, a village in Hungary * Cenad, known in Serbian as Čanad, a commune in Romania See also * Canad Inns, a chain of hotels * Canard (other) * Csanád (other) * Kanad, a town in India * Canada Canada is a country in North America. Its ten provinces and three territories extend from the Atlantic Ocean to the Pacific Ocean and northward into the Arctic Ocean, covering over , making it the world's second-largest country by tot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with also often stylized as or \mathbb. History The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes ''in a fair way'' between two players, who have to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Graph
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs lies at the intersection between graph theory and probability theory. From a mathematical perspective, random graphs are used to answer questions about the properties of ''typical'' graphs. Its practical applications are found in all areas in which complex networks need to be modeled – many random graph models are thus known, mirroring the diverse types of complex networks encountered in different areas. In a mathematical context, ''random graph'' refers almost exclusively to the Erdős–Rényi random graph model. In other contexts, any graph model may be referred to as a ''random graph''. Models A random graph is obtained by starting with a set of ''n'' isolated vertices and adding successive edges between them at random. The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zoltán Füredi
Zoltán Füredi (Budapest, Hungary, 21 May 1954) is a Hungarian mathematician, working in combinatorics, mainly in discrete geometry and extremal combinatorics. He was a student of Gyula O. H. Katona. He is a corresponding member of the Hungarian Academy of Sciences (2004). He is a research professor of the Rényi Mathematical Institute of the Hungarian Academy of Sciences, and a professor at the University of Illinois Urbana-Champaign (UIUC). Füredi received his Candidate of Sciences degree in mathematics in 1981 from the Hungarian Academy of Sciences. Some results * In infinitely many cases he determined the maximum number of edges in a graph with no ''C''4. * With Paul Erdős he proved that for some ''c''>1, there are ''c''''d'' points in ''d''-dimensional space such that all triangles formed from those points are acute. * With Imre Bárány he proved that no polynomial time algorithm determines the volume of convex bodies in dimension ''d'' within a multiplicative ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
