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Graph Labeling
In the mathematical discipline of graph theory, a graph labelling is the assignment of labels, traditionally represented by integers, to edges and/or vertices of a graph. Formally, given a graph , a vertex labelling is a function of to a set of labels; a graph with such a function defined is called a vertex-labeled graph. Likewise, an edge labelling is a function of to a set of labels. In this case, the graph is called an edge-labeled graph. When the edge labels are members of an ordered set (e.g., the real numbers), it may be called a weighted graph. When used without qualification, the term labeled graph generally refers to a vertex-labeled graph with all labels distinct. Such a graph may equivalently be labeled by the consecutive integers , where is the number of vertices in the graph. For many applications, the edges or vertices are given labels that are meaningful in the associated domain. For example, the edges may be assigned weights representing the "cost" of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graceful Labeling
In graph theory, a graceful labeling of a graph with edges is a labeling of its vertices with some subset of the integers from 0 to inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and inclusive. Virginia Vassilevska, "Coding and Graceful Labeling of trees." SURF 2001PostScript/ref> A graph which admits a graceful labeling is called a graceful graph. The name "graceful labeling" is due to Solomon W. Golomb; this type of labeling was originally given the name β-labeling by Alexander Rosa in a 1967 paper on graph labelings.. A major conjecture in graph theory is the graceful tree conjecture or Ringel–Kotzig conjecture, named after Gerhard Ringel and Anton Kotzig, and sometimes abbreviated GTC. It hypothesizes that all trees are graceful. It is still an open conjecture, although a related but weaker conjecture known as "Ringel's conjecture" was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Book (graph Theory)
In graph theory, a book graph (often written B_p ) may be any of several kinds of graph formed by multiple cycles sharing an edge. Variations One kind, which may be called a quadrilateral book, consists of ''p'' quadrilaterals sharing a common edge (known as the "spine" or "base" of the book). That is, it is a Cartesian product of a star and a single edge. The 7-page book graph of this type provides an example of a graph with no harmonious labeling. A second type, which might be called a triangular book, is the complete tripartite graph ''K''1,1,''p''. It is a graph consisting of p triangles sharing a common edge. A book of this type is a split graph. This graph has also been called a K_e(2,p) or a thagomizer graph (after thagomizers, the spiked tails of stegosaurian dinosaurs, because of their pointy appearance in certain drawings) and their graphic matroids have been called thagomizer matroids. Triangular books form one of the key building blocks of line perfect graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Petersen Graph
In the mathematics, mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertex (graph theory), vertices and 15 edge (graph theory), edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest Bridge (graph theory), bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by . Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Odd Cycle Transversal
In graph theory, an odd cycle transversal of an undirected graph is a set of vertices of the graph that has a nonempty intersection with every odd cycle in the graph. Removing the vertices of an odd cycle transversal from a graph leaves a bipartite graph as the remaining induced subgraph. Relation to vertex cover A given n-vertex graph G has an odd cycle transversal of size k, if and only if the Cartesian product of graphs G\square K_2 (a graph consisting of two copies of G, with corresponding vertices of each copy connected by the edges of a perfect matching) has a vertex cover of size n+k. The odd cycle transversal can be transformed into a vertex cover by including both copies of each vertex from the transversal and one copy of each remaining vertex, selected from the two copies according to which side of the bipartition contains it. In the other direction, a vertex cover of G\square K_2 can be transformed into an odd cycle transversal by keeping only the vertices for which bot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Institute Of Combinatorics And Its Applications
The Institute of Combinatorics and its Applications (ICA) is an international scientific organization formed in 1990 to increase the visibility and influence of the combinatorial community. In pursuit of this goal, the ICA sponsors conferences, publishes a bulletin and awards a number of medals, including the Euler, Hall, Kirkman, and Stanton Medals. It is based in Duluth, Minnesota and its operation office is housed at University of Minnesota Duluth. The institute was minimally active between 2010 and 2016 and resumed its full activities in March 2016. Membership The ICA has over 800 members in over forty countries. Membership is at three levels. ''Members'' are those who have not yet completed a Ph.D. ''Associate Fellows'' are younger members who have received the Ph.D. or have published extensively; normally an Associate Fellow should hold the rank of Assistant Professor. ''Fellows'' are expected to be established scholars and typically have the rank of Associate Professor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Anton Kotzig
Anton Kotzig (22 October 1919 – 20 April 1991) was a Slovak–Canadian mathematician, expert in statistics, combinatorics and graph theory. The Ringel–Kotzig conjecture on graceful labeling of trees is named after him and Gerhard Ringel. Kotzig's theorem on the degrees of vertices in convex polyhedra is also named after him. Biography Kotzig was born in Kočovce, a village in Western Slovakia, in 1919. He studied at the secondary grammar school in Nové Mesto nad Váhom, and began his undergraduate studies at Charles University in Prague. After the closure of Czech universities in 1939, he moved to Bratislava, where in 1943 he earned a doctoral degree (RNDr.) in mathematical statistics from Comenius University in Bratislava. He remained in Bratislava working at the Central Bureau of Social Insurance for Slovakia, as the head of department of mathematical statistics. Later he published a book on economy planning. From 1951 to 1959, he lectured at Vysoká škola Ekono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Caterpillar Tree
In graph theory, a caterpillar or caterpillar tree is a tree in which all the vertices are within distance 1 of a central path. Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by Arthur Hobbs. As colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed.". Equivalent characterizations The following characterizations all describe the caterpillar trees: *They are the trees for which removing the leaves and incident edges produces a path graph. *They are the trees in which there exists a path that contains every vertex of degree two or more. *They are the trees in which every vertex of degree at least three has at most two non-leaf neighbors. *They are the trees that do not contain as a subgraph the graph formed by replacing every edge in the star graph ''K''1,3 by a path of length two. *They are the connected graphs that can be drawn with their vertices on two parallel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Path Graph
In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). As Dynkin diagrams In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and the Weyl group o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |