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Arboricity
The arboricity of an undirected graph is the minimum number of forests into which its edges can be partitioned. Equivalently it is the minimum number of spanning forests needed to cover all the edges of the graph. The Nash-Williams theorem provides necessary and sufficient conditions for when a graph is ''k''-arboric. Example The figure shows the complete bipartite graph ''K''4,4, with the colors indicating a partition of its edges into three forests. ''K''4,4 cannot be partitioned into fewer forests, because any forest on its eight vertices has at most seven edges, while the overall graph has sixteen edges, more than double the number of edges in a single forest. Therefore, the arboricity of ''K''4,4 is three. Arboricity as a measure of density The arboricity of a graph is a measure of how dense the graph is: graphs with many edges have high arboricity, and graphs with high arboricity must have a dense subgraph. In more detail, as any n-vertex forest has at most n-1 edges, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (graph Theory)
In graph theory, a star is the complete bipartite graph a tree with one internal node and leaves (but no internal nodes and leaves when ). Alternatively, some authors define to be the tree of order with maximum diameter 2; in which case a star of has leaves. A star with 3 edges is called a claw. The star is edge-graceful when is even and not when is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when ), girth ∞ (it has no cycles), chromatic index , and chromatic number 2 (when ). Additionally, the star has large automorphism group, namely, the symmetric group on letters. Stars may also be described as the only connected graphs in which at most one vertex has degree greater than one. Relation to other graph families Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash-Williams Theorem
In graph theory, the Nash-Williams theorem is a tree-packing theorem that describes how many edge-disjoint spanning trees (and more generally forests) a graph can have:A graph ''G'' has ''t'' edge-disjoint spanning trees iff for every partition V_1, \ldots, V_k \subset V(G) where V_i \neq \emptyset there are at least ''t''(''k'' − 1) crossing edges (Tutte 1961, Nash-Williams 1961).For this article, we will say that such a graph has arboricity ''t'' or is ''t''-arboric. (The actual definition of arboricity is slightly different and applies to forests rather than trees.) Related tree-packing properties A ''k''-arboric graph is necessarily ''k''-edge connected. The converse is not true. As a corollary of NW, every 2''k''-edge connected graph is ''k''-arboric. Both NW and Menger's theorem characterize when a graph has ''k'' edge-disjoint paths between two vertices. Nash-Williams theorem for forests NW (1964) generalized the above result to forests:G can be pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoforest
In graph theory, a pseudoforest is an undirected graphThe kind of undirected graph considered here is often called a multigraph or pseudograph, to distinguish it from a simple graph. in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two cycles of consecutive edges share any vertex with each other, nor can any two cycles be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest. The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan. attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems.. Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests ... [...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] |
Algorithmica
''Algorithmica'' received the highest possible ranking “A*”. External links Springer information Computer science journals Springer Science+Business Media academic journals Monthly journals Publications established in 1986 English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.623. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphs And Combinatorics
''Graphs and Combinatorics'' (ISSN 0911-0119, abbreviated ''Graphs Combin.'') is a peer-reviewed academic journal in graph theory, combinatorics, and discrete geometry published by Springer Japan. Its editor-in-chief is Katsuhiro Ota of Keio University. The journal was first published in 1985. Its founding editor in chief was Hoon Heng Teh of Singapore, the president of the Southeast Asian Mathematics Society, and its managing editor was Jin Akiyama. Originally, it was subtitled "An Asian Journal". In most years since 1999, it has been ranked as a second-quartile journal in discrete mathematics and theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ... by SCImago Journal Rank.. References {{reflist Publications established in 1985 Combinatorics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem (Magnes Press). Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.754. External links * Mathematics journals Publications established in 1963 English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldberg–Seymour Conjecture
In graph theory the Goldberg–Seymour conjecture states that : \chi'(G) \leq \max(\Delta(G) + 1, \Gamma(G)) where \chi'(G) is the edge chromatic number of ''G'' and : \Gamma(G) = \displaystyle\max_ \frac Note this above quantity is twice the arboricity of ''G''. It is sometimes called the density of ''G''. Above ''G'' can be a multigraph (can have loops). Background It is already known that for loopless ''G'' (but can have parallel edges): : \chi'(G) \geq \max\. When does equality not hold? It does not hold for the Petersen graph. It is hard to find other examples. It is currently unknown whether there are any planar graphs for which equality does not hold. This conjecture is named after Mark K. Goldberg of Rensselaer Polytechnic Institute and Paul Seymour of Princeton University, who arrived to it independently of Goldberg. Announced proof In 2019, an alleged proof was announced by Chen, Jing, and Zang in the paper. Part of their proof was to find a suita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(a,b)-decomposability
In graph theory, the (''a'', ''b'')-decomposition of an undirected graph is a partition of its edges into ''a'' + 1 sets, each one of them inducing a forest, except one which induces a graph with maximum degree ''b''. If this graph is also a forest, then we call this a F(''a'', ''b'')-decomposition. A graph with arboricity ''a'' is (''a'', 0)-decomposable. Every (''a'', ''0'')-decomposition or (''a'', ''1'')-decomposition is a F(''a'', ''0'')-decomposition or a F(''a'', ''1'')-decomposition respectively. Graph classes * Every planar graph is F(2, 4)-decomposable. * Every planar graph G with girth at least g is ** F(2, 0)-decomposable if g \ge 4.Implied by . ** (1, 4)-decomposable if g \ge 5. ** F(1, 2)-decomposable if g \ge 6. ** F(1, 1)-decomposable if g \ge 8, or if every cycle of G is either a triangle or a cycle with at least 8 edges not belonging to a triangle. ** (1, 5)-decomposable if G ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strength Of A Graph
In the branch of mathematics called graph theory, the strength of an undirected graph corresponds to the minimum ratio ''edges removed''/''components created'' in a decomposition of the graph in question. It is a method to compute partitions of the set of vertices and detect zones of high concentration of edges, and is analogous to graph toughness which is defined similarly for vertex removal. Definitions The strength \sigma(G) of an undirected simple graph ''G'' = (''V'', ''E'') admits the three following definitions: * Let \Pi be the set of all partitions of V, and \partial \pi be the set of edges crossing over the sets of the partition \pi\in\Pi, then \displaystyle\sigma(G)=\min_\frac. * Also if \mathcal T is the set of all spanning trees of ''G'', then :: \sigma(G)=\max\left\. * And by linear programming duality, :: \sigma(G)=\min\left\. Complexity Computing the strength of a graph can be done in polynomial time, and the first such algorithm was disco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
