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Total Coloring
In graph theory, total coloring is a type of graph coloring on the vertices and edges of a graph. When used without any qualification, a total coloring is always assumed to be ''proper'' in the sense that no adjacent edges, no adjacent vertices and no edge and either endvertex are assigned the same color. The total chromatic number χ″(''G'') of a graph ''G'' is the fewest colors needed in any total coloring of ''G''. The total graph ''T'' = ''T''(''G'') of a graph ''G'' is a graph such that (i) the vertex set of ''T'' corresponds to the vertices and edges of ''G'' and (ii) two vertices are adjacent in ''T'' if and only if their corresponding elements are either adjacent or incident in ''G''. Then total coloring of ''G'' becomes a (proper) vertex coloring of ''T''(''G''). A total coloring is a partitioning of the vertices and edges of the graph into total independent sets. Some inequalities for χ″(''G''): # χ″(''G'') ≥ Δ(''G'') + 1. # χ″(''G'') ≤ Δ(''G'') + 102 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Coloring Foster Cage
Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are comparable). * Total function, a partial function that is also a total relation Business * TotalEnergies, a French petroleum company * Total (cereal), a food brand by General Mills * Total, a brand of strained yogurt made by Fage * Total, a database management system marketed by Cincom Systems * Total Linhas Aéreas - a brazilian airline * Total, a line of dental products by Colgate Music and culture * Total (group), an American R&B girl group * '' Total: From Joy Division to New Order'', a compilation album * ''Total'' (Sebastian album) * ''Total'' (Total album) * ''Total'' (Teenage Bottlerocket album) * ''Total'' (Seigmen album) * ''Total'' (Wanessa album) * ''Total'' (Belinda Peregrín album) * '' Total 1'', an annual compilati ... [...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 combine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Coloring
Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory. It is a generalization of ordinary graph coloring. In a traditional graph coloring, each vertex in a graph is assigned some color, and adjacent vertices — those connected by edges — must be assigned different colors. In a fractional coloring however, a ''set'' of colors is assigned to each vertex of a graph. The requirement about adjacent vertices still holds, so if two vertices are joined by an edge, they must have no colors in common. Fractional graph coloring can be viewed as the linear programming relaxation of traditional graph coloring. Indeed, fractional coloring problems are much more amenable to a linear programming approach than traditional coloring problems. Definitions A ''b''-fold coloring of a graph ''G'' is an assignment of sets of size ''b'' to vertices of a graph such that adjacent vertices receive disjoint sets. An ''a'':''b''-coloring is a ''b''-fold colori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the 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 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 additional assumptions such as the absence of isthmuses, is called a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a 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 cycles. The two sets U and V may be thought of as a 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 triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vadim G
Vadim (Cyrillic: Вадим) is a Russian, Ukrainian, Romanian, Slovene masculine given name derived either from the Persian ''badian'' (anise or aniseed), or from the Ruthenian word ''volod'' (russian: волод), meaning ''to rule'' or ''vaditi'' (russian: вадити), meaning ''to blame''. Its long version, Vadimir, is now obsolete. Dictionary of Russian Names This given name is highly popular in (as Vadim), (as ) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mehdi Behzad
Mehdi Behzad (Persian:مهدی بهزاد; born April 22, 1936) is an Iranian mathematician specializing in graph theory. He introduced his total coloring theory (also known as "Behzad's conjecture" or "the total chromatic number conjecture") during his Ph.D. studies in 1965. Despite the active work during the last 50 years this conjecture remains as challenging as it is open. In fact, Behzad's conjecture now belongs to mathematics’ classic open problems. Behzad has been instrumental in institutionalizing mathematics education and popularization of mathematics in Iran, and has received numerous awards and recognition for his lifetime service to the Iranian scientific community. Graph theory Behzad is the coauthor of two text books on graph theory published in 1972 and 1979 in the U.S., which were among the key references on this new field of mathematics. He has been one of the direct collaborators of Paul Erdős. Professorship Behzad was the first faculty member of Sharif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Coloring
In graph theory, an 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 number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Edge-coloring
In mathematics, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring. An instance of a list edge-coloring problem consists of a graph together with a list of allowed colors for each edge. A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper if no two adjacent edges receive the same color. A graph ''G'' is ''k''-edge-choosable if every instance of list edge-coloring that has ''G'' as its underlying graph and that provides at least ''k'' allowed colors for each edge of ''G'' has a proper coloring. The edge choosability, or ''list edge colorability'', ''list edge chromatic number'', or ''list chromatic index'', ch′(''G'') of graph ''G'' is the least number ''k'' such that ''G'' is ''k''-edge-choosable. It is conjectured that it always equals the chromatic index. Properties Some properties of ch′(''G''): # ch′(''G'') < 2 χ′(''G''). # ch′(K''n'',''n'' ... [...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 wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Graph Theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Independent Set
Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are comparable). * Total function, a partial function that is also a total relation Business * TotalEnergies, a French petroleum company * Total (cereal), a food brand by General Mills * Total, a brand of strained yogurt made by Fage * Total, a database management system marketed by Cincom Systems * Total Linhas Aéreas - a brazilian airline * Total, a line of dental products by Colgate Music and culture * Total (group), an American R&B girl group * '' Total: From Joy Division to New Order'', a compilation album * ''Total'' (Sebastian album) * ''Total'' (Total album) * ''Total'' (Teenage Bottlerocket album) * ''Total'' (Seigmen album) * ''Total'' (Wanessa album) * ''Total'' (Belinda Peregrín album) * '' Total 1'', an annual compilati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
