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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] |
Mathematics
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] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Coloring
In graph theory, a branch of mathematics, list coloring is a type of graph coloring where each vertex can be restricted to a list of allowed colors. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. Definition Given a graph ''G'' and given a set ''L''(''v'') of colors for each vertex ''v'' (called a list), a list coloring is a ''choice function'' that maps every vertex ''v'' to a color in the list ''L''(''v''). As with graph coloring, a list coloring is generally assumed to be proper, meaning no two adjacent vertices receive the same color. A graph is ''k''-choosable (or ''k''-list-colorable) if it has a proper list coloring no matter how one assigns a list of ''k'' colors to each vertex. The choosability (or list colorability or list chromatic number) ch(''G'') of a graph ''G'' is the least number ''k'' such that ''G'' is ''k''-choosable. More generally, for a function ''f'' assigning a positive integer ''f''(''v'') to each v ... [...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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Index
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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dinitz Conjecture
In combinatorics, the Dinitz theorem (formerly known as Dinitz conjecture) is a statement about the extension of arrays to partial Latin squares, proposed in 1979 by Jeff Dinitz, and proved in 1994 by Fred Galvin. The Dinitz theorem is that given an ''n'' × ''n'' square array, a set of ''m'' symbols with ''m'' ≥ ''n'', and for each cell of the array an ''n''-element set drawn from the pool of ''m'' symbols, it is possible to choose a way of labeling each cell with one of those elements in such a way that no row or column repeats a symbol. It can also be formulated as a result in graph theory, that the list chromatic index of the complete bipartite graph K_ equals n. That is, if each edge of the complete bipartite graph is assigned a set of n colors, it is possible to choose one of the assigned colors for each edge such that no two edges incident to the same vertex have the same color. Galvin's proof generalizes to the statement that, for every bipartite multigraph, the lis ... [...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 of ... [...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 w ... [...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] |
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, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
