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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 property, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turan 13-4
Turan (; ; , , ) is a historical region in Central Asia. The term is of Iranian origin and may refer to a particular prehistoric human settlement, a historic geographical region, or a culture. The original Turanians were an Iranian tribe of the Avestan age. Overview In ancient Iranian mythology, Tūr or Turaj (''Tuzh'' in Middle Persian) is the son of the emperor Fereydun. According to the account in the ''Shahnameh'', the nomadic tribes who inhabited these lands were ruled by Tūr. In that sense, the Turanians could be members of two Iranian peoples both descending from Fereydun, but with different geographical domains and often at war with each other. Turan, therefore, comprised five areas: the Kopet Dag region, the Atrek valley, parts of Bactria, Sogdia and Margiana. A later association of the original Turanians with Turkic peoples is based primarily on the subsequent Turkification of Central Asia, including the above areas. According to C. E. Bosworth, however ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique Number
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphon
In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function W: ,12\to ,1/math>, that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable random graph models. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary large dense graphs. Statistical formulation A graphon is a symmetric measurable function W: ,12 \to ,1/math>. Usually a graphon is understood as defining an exchangeable random graph model according to the following scheme: # Each vertex j of the graph is assigned an independent random value u_j\sim U ,1/math> # Edge (i,j) is independently included in the graph with probability W(u_i,u_j). A random graph model is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Homomorphism
In the mathematics, mathematical field of graph theory, a graph homomorphism is a mapping between two graph (discrete mathematics), graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertex (graph theory), vertices to adjacent vertices. Homomorphisms generalize various notions of graph colorings and allow the expression of an important class of constraint satisfaction problems, such as certain Scheduling (production processes), scheduling or frequency assignment problems. The fact that homomorphisms can be composed leads to rich algebraic structures: a preorder on graphs, a distributive lattice, and a category (mathematics), category (one for undirected graphs and one for directed graphs). The computational complexity of finding a homomorphism between given graphs is prohibitive in general, but a lot is known about special cases that are solvable in Time complexity#Polynomial time, polynomial time. Boun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 graph, 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 vertex (graph theory), vertices U and V, and a set of edge (graph theory), 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_. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), 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 cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, 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 Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán Graph
The Turán graph, denoted by T(n,r), is a complete multipartite graph; it is formed by partitioning a set of n vertices into r subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to different subsets. Where q and s are the quotient and remainder of dividing n by r (so n = qr + s), the graph is of the form K_, and the number of edges is : \left(1 - \frac\right)\frac + . For r\le7, this edge count can be more succinctly stated as \left\lfloor\left(1-\frac1r\right)\frac2\right\rfloor. The graph has s subsets of size q+ 1 , and r - s subsets of size q; each vertex has degree n-q-1 or n-q. It is a regular graph if n is divisible by r (i.e. when s=0). Turán's theorem Turán graphs are named after Pál Turán, who used them to prove Turán's theorem, an important result in extremal graph theory. By the pigeonhole principle, every set of ''r'' + 1 vertices in the Turán graph includes two vertices in the same part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vizing's Theorem
In graph theory, Vizing's theorem states that every simple undirected graph may be edge colored using a number of colors that is at most one larger than the maximum degree of the graph. At least colors are always necessary, so the undirected graphs may be partitioned into two classes: "class one" graphs for which colors suffice, and "class two" graphs for which colors are necessary. A more general version of Vizing's theorem states that every undirected multigraph without loops can be colored with at most colors, where is the multiplicity of the multigraph. The theorem is named for Vadim G. Vizing who published it in 1964. Discovery The theorem discovered by Soviet mathematician Vadim G. Vizing was published in 1964 when Vizing was working in Novosibirsk and became known as Vizing's theorem. Indian mathematician R. P. Gupta independently discovered the theorem, while undertaking his doctorate (1965-1967). Examples When , the graph must itself be a matching, with no two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Coloring
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-color Theorem
In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions share a common boundary of non-zero length (i.e., not merely a corner where three or more regions meet). It was the first major theorem to be proved using a computer. Initially, this proof was not accepted by all mathematicians because the computer-assisted proof was infeasible for a human to check by hand. The proof has gained wide acceptance since then, although some doubts remain. The theorem is a stronger version of the five color theorem, which can be shown using a significantly simpler argument. Although the weaker five color theorem was proven already in the 1800s, the four color theorem resisted until 1976 when it was proven by Kenneth Appel and Wolfgang Haken in a computer-aided proof. This came after many false proofs and mistak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
