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Blow-up Lemma
The blow-up lemma, proved by János Komlós, Gábor N. Sárközy, and Endre Szemerédi in 1997, is an important result in extremal graph theory, particularly within the context of the regularity method. It states that the regular pairs in the statement of Szemerédi's regularity lemma behave like complete bipartite graphs in the context of embedding spanning graphs of bounded degree. Definitions and Statement To formally state the blow-up lemma, we first need to define the notion of a super-regular pair. Super-regular pairs A pair (A,B) of subsets of the vertex set is called (\varepsilon, \delta)-super-regular if for every X \subset A and Y \subset B satistying : , X, > \varepsilon , A, and , Y, > \varepsilon , B, we have : e(X,Y) > \delta , X, , Y, and furthermore, : \deg(a) > \delta , B, for all a \in A and \deg(b) > \delta , A, for all b \in B . Here e(X, Y) denotes the number of pairs (x,y) with x \in X and y \in Y such that \ is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
János Komlós (mathematician)
János Komlós (born 23 May 1942, in Budapest) is a Hungarian-American mathematician, working in probability theory and discrete mathematics. He has been a professor of mathematics at Rutgers University since 1988. He graduated from the Eötvös Loránd University, then became a fellow at the Mathematical Institute of the Hungarian Academy of Sciences. Between 1984–1988 he worked at the University of California, San Diego. Notable results * He proved that every L1-bounded sequence of real functions contains a subsequence such that the arithmetic means of all its subsequences converge pointwise almost everywhere. In probabilistic terminology, the theorem is as follows. Let ξ1,ξ2,... be a sequence of random variables such that ''E'' 1''E'' 2... is bounded. Then there exist a subsequence ξ'1, ξ'2,... and a random variable β such that for each further subsequence η1,η2,... of ξ'0, ξ'1,... we have (η1+...+ηn)/n → β a.s. * With Miklós Ajtai and Endre Szemeréd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Power
In graph theory, a branch of mathematics, the th power of an undirected graph is another graph that has the same set of vertices, but in which two vertices are adjacent when their distance in is at most . Powers of graphs are referred to using terminology similar to that of exponentiation of numbers: is called the ''square'' of , is called the ''cube'' of , etc. Graph powers should be distinguished from the products of a graph with itself, which (unlike powers) generally have many more vertices than the original graph. Properties If a graph has diameter , then its -th power is the complete graph. If a graph family has bounded clique-width, then so do its -th powers for any fixed . Coloring Graph coloring on the square of a graph may be used to assign frequencies to the participants of wireless communication networks so that no two participants interfere with each other at any of their common neighbors, and to find graph drawings with high angular resolution. Both the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandom Graph
In graph theory, a graph is said to be a pseudorandom graph if it obeys certain properties that random graphs obey with high probability. There is no concrete definition of graph pseudorandomness, but there are many reasonable characterizations of pseudorandomness one can consider. Pseudorandom properties were first formally considered by Andrew Thomason in 1987. He defined a condition called "jumbledness": a graph G=(V,E) is said to be (p,\alpha)-''jumbled'' for real p and \alpha with 0 [...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 aim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sparse Graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and depends on context. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of edges for an undirected graph is = \frac2, so the maximal density is 1 (for complete graphs) and the minimal density is 0 . Upper density ''Upper density'' is an extension of the concept of g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rainbow Coloring
In graph theory, a path in an edge-colored graph is said to be rainbow if no color repeats on it. A graph is said to be rainbow-connected (or rainbow colored) if there is a rainbow path between each pair of its vertices. If there is a rainbow shortest path between each pair of vertices, the graph is said to be strongly rainbow-connected (or strongly rainbow colored). Definitions and bounds The rainbow connection number of a graph G is the minimum number of colors needed to rainbow-connect G, and is denoted by \text(G). Similarly, the strong rainbow connection number of a graph G is the minimum number of colors needed to strongly rainbow-connect G, and is denoted by \text(G). Clearly, each strong rainbow coloring is also a rainbow coloring, while the converse is not true in general. It is easy to observe that to rainbow-connect any connected graph G, we need at least \text(G) colors, where \text(G) is the diameter of G (i.e. the length of the longest shortest path). On th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Keevash
Peter Keevash (born 30 November 1978) is a British mathematician, working in combinatorics. He is Professor of Mathematics at the University of Oxford and a Fellow of Mansfield College. Early years Keevash was born in Brighton, England, but mostly grew up in Leeds. He competed in the International Mathematical Olympiad in 1995. He entered Trinity College, University of Cambridge, in 1995 and completed his B.A. in mathematics in 1998. He earned his doctorate from Princeton University with Benny Sudakov as advisor. He took a postdoctoral position at the California Institute of Technology before moving to Queen Mary, University of London as a lecturer, and subsequently professor, before his move to Oxford in September 2013. Mathematics Keevash has published many results in combinatorics, particularly in extremal graph and hypergraph theory and Ramsey Theory. In joint work with Tom Bohman he established the best-known lower bound for the off-diagonal Ramsey Number R(3,k) , na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Factorization
In graph theory, a factor of a graph ''G'' is a spanning subgraph, i.e., a subgraph that has the same vertex set as ''G''. A ''k''-factor of a graph is a spanning ''k''- regular subgraph, and a ''k''-factorization partitions the edges of the graph into disjoint ''k''-factors. A graph ''G'' is said to be ''k''-factorable if it admits a ''k''-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a ''k''-regular graph is an edge coloring with ''k'' colors. A 2-factor is a collection of cycles that spans all vertices of the graph. 1-factorization If a graph is 1-factorable (ie, has a 1-factorization), then it has to be a regular graph. However, not all regular graphs are 1-factorable. A ''k''-regular graph is 1-factorable if it has chromatic index ''k''; examples of such graphs include: * Any regular bipartite graph. Hall's marriage theorem can be used to show that a ''k''-regular bipartite graph contains a perfect matching. One can then remove t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equitable Coloring
In graph theory, an area of mathematics, an equitable coloring is an assignment of colors to the vertices of an undirected graph, in such a way that *No two adjacent vertices have the same color, and *The numbers of vertices in any two color classes differ by at most one. That is, the partition of vertices among the different colors is as uniform as possible. For instance, giving each vertex a distinct color would be equitable, but would typically use many more colors than are necessary in an optimal equitable coloring. An equivalent way of defining an equitable coloring is that it is an embedding of the given graph as a subgraph of a Turán graph with the same set of vertices. There are two kinds of chromatic number associated with equitable coloring.. The equitable chromatic number of a graph ''G'' is the smallest number ''k'' such that ''G'' has an equitable coloring with ''k'' colors. But ''G'' might not have equitable colorings for some larger numbers of colors; the equitable c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raphael Yuster
Raffaello Sanzio da Urbino, better known as Raphael (; or ; March 28 or April 6, 1483April 6, 1520), was an Italian painter and architect of the High Renaissance. His work is admired for its clarity of form, ease of composition, and visual achievement of the Neoplatonic ideal of human grandeur. Together with Leonardo da Vinci and Michelangelo, he forms the traditional trinity of great masters of that period. His father was court painter to the ruler of the small but highly cultured city of Urbino. He died when Raphael was eleven, and Raphael seems to have played a role in managing the family workshop from this point. He trained in the workshop of Perugino, and was described as a fully trained "master" by 1500. He worked in or for several cities in north Italy until in 1508 he moved to Rome at the invitation of the pope, to work on the Vatican Palace. He was given a series of important commissions there and elsewhere in the city, and began to work as an architect. He was sti ... [...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] |
Paul Seymour (mathematician)
Paul D. Seymour (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory. He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website. Seymour is currently the Albert Baldwin Dod Professor of Mathematics at Princeton University. He won a Sloan Fellowship in 1983, and the Ostrowski Prize in 2004; and (sometimes with others) won the Fulkerson Prize in 1979, 1994, 2006 and 2009, and the Pólya Prize in 1983 and 2004. He received an honorary doctorate from the University of Waterloo in 2008, one from the Technical University of Denmark in 2013, and one from the École normale supérieure de Lyon in 2022. He was an invited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
