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Rainbow Matching
In the mathematical discipline of graph theory, a rainbow matching in an Edge coloring, edge-colored graph is a Matching (graph theory), matching in which all the edges have distinct colors. Definition Given an edge-colored graph , a rainbow matching in is a set of pairwise non-adjacent edges, that is, no two edges share a common vertex, such that all the edges in the set have distinct colors. A maximum rainbow matching is a rainbow matching that contains the largest possible number of edges. History Rainbow matchings are of particular interest given their connection to transversals of Latin squares. Denote by the complete bipartite graph on vertices. Every proper -edge coloring of corresponds to a Latin square of order . A rainbow matching then corresponds to a Latin square#Transversals and rainbow matchings, transversal of the Latin square, meaning a selection of positions, one in each row and each column, containing distinct entries. This connection between transversa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Conical Hull
Given a finite number of vectors x_1, x_2, \dots, x_n in a real vector space, a conical combination, conical sum, or weighted sum''Convex Analysis and Minimization Algorithms'' by Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, 1993, pp. 101, 102/ref>''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> of these vectors is a vector of the form : \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where \alpha_i are non-negative real numbers. The name derives from the fact that the set of all conical sum of vectors defines a cone (possibly in a lower-dimensional subspace). Conical hull The set of all conical combinations for a given set ''S'' is called the conical hull of ''S'' and denoted ''cone''(''S'') or ''coni''(''S''). That is, :\operatorname (S)=\left\. By taking ''k'' = 0, it follows the zero vector (origin) belongs to all conical hulls (since the summation becomes an empty sum). The conical hull of a set ''S'' is a convex set. In fact, it is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rainbow Covering
In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure 'covers' another, or how large the structure has to be to do that. Covering problems are Optimization (mathematics), minimization problems and usually integer linear programs, whose dual problems are called packing problems. The most prominent examples of covering problems are the set cover problem, which is equivalent to the Hitting set, hitting set problem, and its special cases, the vertex cover problem and the edge cover problem. Covering problems allow the covering primitives to overlap; the process of covering something with non-overlapping primitives is called decomposition (other), decomposition. General linear programming formulation In the context of linear programming, one can think of any minimization linear program as a covering problem if the coefficients in the constraint matrix (mathematics), matrix, the objective function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rainbow-independent Set
In graph theory, a rainbow-independent set (ISR) is an independent set in a graph, in which each vertex has a different color. Formally, let be a graph, and suppose vertex set is partitioned into subsets , called "colors". A set of vertices is called a rainbow-independent set if it satisfies both the following conditions: * It is an independent set – every two vertices in are not adjacent (there is no edge between them); * It is a rainbow set – contains at most a single vertex from each color . Other terms used in the literature are independent set of representatives, independent transversal, and independent system of representatives. As an example application, consider a faculty with departments, where some faculty members dislike each other. The dean wants to construct a committee with members, one member per department, but without any pair of members who dislike each other. This problem can be presented as finding an ISR in a graph in which the nodes are the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rainbow-colorable 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 hyperg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
