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Hall-type Theorems For Hypergraphs
In the mathematical field of graph theory, Hall-type theorems for hypergraphs are several generalizations of Hall's marriage theorem from graphs to hypergraphs. Such theorems were proved by Ofra Kessler, Ron Aharoni, Penny Haxell, Roy Meshulam, and others. Preliminaries Hall's marriage theorem provides a condition guaranteeing that a bipartite graph admits a perfect matching, or - more generally - a matching that saturates all vertices of . The condition involves the number of neighbors of subsets of . Generalizing Hall's theorem to hypergraphs requires a generalization of the concepts of bipartiteness, perfect matching, and neighbors. 1. Bipartiteness: The notion of a bipartiteness can be extended to hypergraphs in many ways (see bipartite hypergraph). Here we define a hypergraph as bipartite if it is ''exactly 2- colorable'', i.e., its vertices can be 2-colored such that each hyperedge contains exactly one yellow vertex. In other words, can be partitioned into two se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Hypergraph
In graph theory, the term bipartite hypergraph describes several related classes of hypergraphs, all of which are natural generalizations of a bipartite graph. Property B and 2-colorability The weakest definition of bipartiteness is also called 2-colorability. A hypergraph ''H'' = (''V'', ''E'') is called 2-colorable if its vertex set ''V'' can be partitioned into two sets, ''X'' and ''Y'', such that each hyperedge meets both ''X'' and ''Y''. Equivalently, the vertices of ''H'' can be 2-colored so that no hyperedge is monochromatic. Every bipartite graph ''G'' = (''X''+''Y'', ''E'') is 2-colorable: each edge contains exactly one vertex of ''X'' and one vertex of ''Y'', so e.g. ''X'' can be colored blue and ''Y'' can be colored yellow and no edge is monochromatic. The property of 2-colorability was first introduced by Felix Bernstein in the context of set families; therefore it is also called Property B. Exact 2-colorability A stronger definition of bipartiteness is: a hyper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meshulam's Game
In graph theory, Meshulam's game is a game used to explain a theorem of Roy Meshulam related to the homological connectivity of the independence complex of a graph, which is the smallest index ''k'' such that all reduced homological groups up to and including ''k'' are trivial. The formulation of this theorem as a game is due to Aharoni, Berger and Ziv. Description The game-board is a graph ''G.'' It is a zero-sum game for two players, CON and NON. CON wants to show that I(''G''), the independence complex of ''G'', has a high connectivity; NON wants to prove the opposite. At his turn, CON chooses an edge ''e'' from the remaining graph. NON then chooses one of two options: * ''Disconnection'' – remove the edge ''e'' from the graph. * ''Explosion'' – remove both endpoints of ''e'', together with all their neighbors and the edges incident to them. The score of CON is defined as follows: * If at some point the remaining graph has an isolated vertex, the score is infinity; * O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homological Connectivity
In algebraic topology, homological connectivity is a property describing a topological space based on its homology groups. Definitions Background ''X'' is ''homologically-connected'' if its 0-th homology group equals Z, i.e. H_0(X)\cong \mathbb, or equivalently, its 0-th reduced homology group is trivial: \tilde(X)\cong 0. * For example, when ''X'' is a graph and its set of connected components is ''C'', H_0(X)\cong \mathbb^ and \tilde(X)\cong \mathbb^ (see graph homology). Therefore, homological connectivity is equivalent to the graph having a single connected component, which is equivalent to graph connectivity. It is similar to the notion of a connected space. ''X'' is ''homologically 1-connected'' if it is homologically-connected, and additionally, its 1-th homology group is trivial, i.e. H_1(X)\cong 0. * For example, when ''X'' is a connected graph with vertex-set ''V'' and edge-set ''E'', H_1(X) \cong \mathbb^. Therefore, homological 1-connectivity is equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transversal (combinatorics)
In mathematics, particularly in combinatorics, given a family of sets, here called a collection ''C'', a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually disjoint, each element of the transversal corresponds to exactly one member of ''C'' (the set it is a member of). If the original sets are not disjoint, there are two possibilities for the definition of a transversal: * One variation is that there is a bijection ''f'' from the transversal to ''C'' such that ''x'' is an element of ''f''(''x'') for each ''x'' in the transversal. In this case, the transversal is also called a system of distinct representatives (SDR). * The other, less commonly used, does not require a one-to-one relation between the elements of the transversal and the sets of ''C''. In this situation, the members of the system of representatives are not necessarily distinct. In computer science, comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ryser's Conjecture
In graph theory, Ryser's conjecture is a conjecture relating the maximum matching size and the minimum transversal size in hypergraphs. This conjecture first appeared in 1971 in the Ph.D. thesis of J. R. Henderson, whose advisor was Herbert John Ryser. Preliminaries A matching in a hypergraph is a set of hyperedges such that each vertex appears in at most one of them. The largest size of a matching in a hypergraph ''H'' is denoted by \nu(H). A transversal (or vertex cover) in a hypergraph is a set of vertices such that each hyperedge contains at least one of them. The smallest size of a transversal in a hypergraph ''H'' is denoted by \tau(H). For every ''H'', \nu(H)\leq \tau(H), since every cover must contain at least one point from each edge in any matching. If H is ''r''-uniform (each hyperedge has exactly ''r'' vertices), then \tau(H) \leq r\cdot \nu(H), since the union of the edges from any maximal matching is a set of at most ''rv'' vertices that meets every edge. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sperner's Lemma
In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an simplex contains a cell whose vertices all have different colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms. Finding a Sperner coloring or equivalently a Brouwer fixed point is now believed to be an intractable computational problem, even in the plane, in the general case. The problem is PPAD-complete, a complexity class invented by Christos Papadimitriou. According to the Soviet ''Mathematical Encyclopaedia'' (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Width Of A Hypergraph
In graph theory, there are two related properties of a hypergraph that are called its "width". Given a hypergraph ''H'' = (''V'', ''E''), we say that a set ''K'' of edges ''pins'' another set ''F'' of edges if every edge in ''F'' intersects some edge in ''K''. Then: * The width of ''H'', denoted w(''H''), is the smallest size of a subset of ''E'' that pins ''E''. * The matching width of ''H'', denoted mw(''H''), is the maximum, over all matchings ''M'' in ''H'', of the minimum size of a subset of ''E'' that pins ''M''. Since ''E'' contains all matchings in ''E'', for all ''H'': w(''H'') ≥ mw(''H''). The width of a hypergraph is used in Hall-type theorems for hypergraphs. Examples Let ''H'' be the hypergraph with vertex set V = and edge set:''E'' = The widths of ''H'' are: * w(''H'') = 2, since ''E'' is pinned e.g. by the set , and cannot be pinned by any smaller set. * mw(''H'') = 1, since every matching can be pinned by a single edge. There are two matchings: is pinned ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fair Item Allocation
Fair item allocation is a kind of a fair division problem in which the items to divide are ''discrete'' rather than continuous. The items have to be divided among several partners who value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios: * Several heirs want to divide the inherited property, which contains e.g. a house, a car, a piano and several paintings. * Several lecturers want to divide the courses given in their faculty. Each lecturer can teach one or more whole courses. *White elephant gift exchange parties The indivisibility of the items implies that a fair division may not be possible. As an extreme example, if there is only a single item (e.g. a house), it must be given to a single partner, but this is not fair to the other partners. This is in contrast to the fair cake-cutting problem, where the dividend is divisible and a fair division always exists. In some cases, the indivisibility pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton (mathematics)
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as \ is a singleton as it contains a single element (which itself is a set, however, not a singleton). A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton \. In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of \, which is the same a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
