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Well-colored Graph
In graph theory, a subfield of mathematics, a well-colored graph is an undirected graph for which greedy coloring uses the same number of colors regardless of the order in which colors are chosen for its vertices. That is, for these graphs, the chromatic number (minimum number of colors) and Grundy number (maximum number of greedily-chosen colors) are equal. Examples The well-colored graphs include the complete graphs and odd-length cycle graphs (the graphs that form the exceptional cases to Brooks' theorem) as well as the complete bipartite graphs and complete multipartite graphs. The simplest example of a graph that is not well-colored is a four-vertex path. Coloring the vertices in path order uses two colors, the optimum for this graph. However, coloring the ends of the path first (using the same color for each end) causes the greedy coloring algorithm to use three colors for this graph. Because there exists a non-optimal vertex ordering, the path is not well-colored. Complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Tripartite Graph Octahedron
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or more associated polypeptide chains * Spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Multipartite Graph
In graph theory, a part of mathematics, a -partite graph is a graph whose vertices are (or can be) partitioned into different independent sets. Equivalently, it is a graph that can be colored with colors, so that no two endpoints of an edge have the same color. When these are the bipartite graphs, and when they are called the tripartite graphs. Bipartite graphs may be recognized in polynomial time but, for any it is NP-complete, given an uncolored graph, to test whether it is -partite. However, in some applications of graph theory, a -partite graph may be given as input to a computation with its coloring already determined; this can happen when the sets of vertices in the graph represent different types of objects. For instance, folksonomies have been modeled mathematically by tripartite graphs in which the three sets of vertices in the graph represent users of a system, resources that the users are tagging, and tags that the users have applied to the resources. A complet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, |
Cograph
In graph theory, a cograph, or complement-reducible graph, or ''P''4-free graph, is a graph that can be generated from the single-vertex graph ''K''1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes ''K''1 and is closed under complementation and disjoint union. Cographs have been discovered independently by several authors since the 1970s; early references include , , , and . They have also been called D*-graphs, hereditary Dacey graphs (after the related work of James C. Dacey Jr. on orthomodular lattices), and 2-parity graphs. They have a simple structural decomposition involving disjoint union and complement graph operations that can be represented concisely by a labeled tree, and used algorithmically to efficiently solve many problems such as finding the maximum clique that are hard on more general graph classes. Special cases of the cographs include the complete graphs, complete bipartite graphs, cluster ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subset V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. * Induced paths are induced subgraphs that are paths. The shortest path betwe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hereditary Property
In mathematics, a hereditary property is a property of an object that is inherited by all of its subobjects, where the meaning of ''subobject'' depends on the context. These properties are particularly considered in topology and graph theory, but also in set theory. In topology In topology, a topological property is said to be ''hereditary'' if whenever a topological space has that property, then so does every Subspace topology, subspace of it. If the latter is true only for Closed set, closed subspaces, then the property is called ''weakly hereditary'' or ''closed-hereditary''. For example, second countability and metrisability are hereditary properties. sequential space, Sequentiality and Compact space, Hausdorff compactness are weakly hereditary, but not hereditary. Connected space, Connectivity is not weakly hereditary. If ''P'' is a property of a topological space ''X'' and every subspace also has property ''P'', then ''X'' is said to be "hereditarily ''P''". In combinatoric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-complete
In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying all possible solutions. # the problem can be used to simulate every other problem for which we can verify quickly that a solution is correct. In this sense, NP-complete problems are the hardest of the problems to which solutions can be verified quickly. If we could find solutions of some NP-complete problem quickly, we could quickly find the solutions of every other problem to which a given solution can be easily verified. The name "NP-complete" is short for "nondeterministic polynomial-time complete". In this name, "nondeterministic" refers to nondeterministic Turing machines, a way of mathematically formalizing the idea of a brute-force search algorithm. Polynomial time refers to an amount of time that is considered "quick" for a det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP (complexity)
In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine, or alternatively the set of problems that can be solved in polynomial time by a nondeterministic Turing machine.''Polynomial time'' refers to how quickly the number of operations needed by an algorithm, relative to the size of the problem, grows. It is therefore a measure of efficiency of an algorithm. An equivalent definition of NP is the set of decision problems ''solvable'' in polynomial time by a nondeterministic Turing machine. This definition is the basis for the abbreviation NP; "nondeterministic, polynomial time". These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by ''edges'' (also called ''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 of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brooks' Theorem
In graph theory, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs and cycle graphs of odd length, which require Δ + 1 colors. The theorem is named after R. Leonard Brooks, who published a proof of it in 1941. A coloring with the number of colors described by Brooks' theorem is sometimes called a ''Brooks coloring'' or a Δ-''coloring''. Formal statement For any connected undirected graph ''G'' with maximum degree Δ, the chromatic number of ''G'' is at most Δ, unless ''G'' is a complete graph or an odd cycle, in which case the chromatic number is Δ + 1. Proof gives a simplified proof of Brooks' theorem. If the graph is not biconnected, its biconnected components may be colored separately and then the colorings combined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
