Eulerian path

An Eulerian path, Eulerian trail or Euler walk in an undirected graph is a path that uses each edge exactly once. If such a path exists, the graph is called traversable. An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. For directed graphs path has to be replaced with directed path and cycle with directed cycle. The definition and properties of Eulerian paths, cycles a ...

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Eulerian path, Eulerian path - Definition, Eulerian path - Notes, Eulerian path - Properties, Eulerian path - Constructing Eulerian paths and cycles, Eulerian path - Counting Eulerian circuits in digraphs

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The best characterization of hamiltonian graphs was given in 1972 by the Bondy-Chvátal theorem which generalizes earlier results by G. A. Dirac and Oystein Ore. It basically states that a graph is hamiltonian if enough edges exist. First we have to define the closure of a graph. Given a graph G with n vertices, the closure cl(G) is uniquely constructed from G by adding for all nonadjacent pairs of vertices u and v with degree(v) + degree(u) ≥ n the new edge uv. Bondy-Chvátal theorem (1972) A graph is ha ...

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Hamiltonian path, Hamiltonian path - Definition, Hamiltonian path - Examples, Hamiltonian path - Notes, Hamiltonian path - Bondy-Chvátal theorem

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Moffatt's work inspired several other workers to investigate the dissipative mechanism of Euler's disk. In the 30 November 2000 issue of Nature, physicists Van den Engh and coworkers discuss experiments in which coins were spun in a vacuum. They found that slippage between the coin and the surface could account for observations, and the presence or absence of air affected the coin's behaviour only slightly. They pointed out that Moffatt's analysis would predict a very ...

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Euler's disk, Euler's disk - Rebuttals

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Leonhard Euler was born near Basel, Switzerland, the son of Paul Euler, a Lutheran minister. Although in his childhood he exhibited great mathematical talents, his father wanted him to study theology and become a minister. In 1720 Euler began his studies at the University of Basel. There Euler met Daniel and Nikolaus Bernoulli, who noticed Euler's skills in mathematics. Paul Euler had attended Jakob Bernoulli's mathematical lectures and respected his family. When Daniel and Nikolaus Bernoulli asked him to allow his son to study mathematics he finally ...

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Leonhard Euler, Leonhard Euler - Biography, Leonhard Euler - Discoveries, Leonhard Euler - Honours, Leonhard Euler - Quotes

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In Greek mythology, the Labyrinth was an elaborate maze-like structure constructed for King Minos of Crete and designed by the legendary artificer Daedalus to hold the Minotaur, a creature that was half man and half bull and was eventually killed by the Athenian hero Theseus. Theseus was aided by Ariadne, who provided him with a fateful thread to wind his way back again. The term labyrinth is often used interchangeably with maze, but a maze is a tour puzzle in the form of a complex branching passage, with choices ...

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Sacred geometry is geometry that is sacred to the observer or discoverer of the geometry. This meaning is sometimes described as being the language of the God of the religion of the people who discovered or used it. Sacred geometry can be described as attributing a religious or cultural value to the graphical representation of the mathematical relationships and the design of the man-made objects that sy ...

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Euler, with Daniel Bernoulli, established the law that the torque on a thin elastic beam is proportional to a measure of the elasticity of the material and the second moment of area of a cross section, about an axis through the center of mass and perpendicular to the plane of the moment, see Euler-Bernoulli beam equation. He also deduced the Euler equations, a set of laws of motion in fluid dynamics, directly from Newton's laws of motion. These equations are formally identical to the Navier-Stokes equations with zero viscosity. They are interesting ch ...

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Leonhard Euler, Leonhard Euler - Biography, Leonhard Euler - Discoveries, Leonhard Euler - Honours, Leonhard Euler - Quotes

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Any Hamiltonian cycle can be converted to a Hamiltonian path by removing one of its edges, but a Hamiltonian path can be extended to Hamiltonian cycle only if its endpoints are adjacent. The line graph of a hamiltonian graph is hamiltonian. The line graph of an eulerian graph is hamiltonian. A tournament (with more than 2 vertices) is hamiltonian if and only if it is strongly connected. ...

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Hamiltonian path, Hamiltonian path - Definition, Hamiltonian path - Examples, Hamiltonian path - Notes, Hamiltonian path - Bondy-Chvátal theorem

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A Hamiltonian path or traceable path is a path that visits each vertex exactly once. A graph that contains a Hamiltonian path is called a traceable graph. A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (excluding the start/end vertex). A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. Similar notions may be defined for directed graphs, where edges (arcs) of a path or a cycle are required to point in th ...

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Hamiltonian path, Hamiltonian path - Definition, Hamiltonian path - Examples, Hamiltonian path - Notes, Hamiltonian path - Bondy-Chvátal theorem

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A graph G consists of two types of elements, namely vertices and edges that are said to connect pairs of vertices. The set of edges is usually defined as a set of two-element subsets of the set of vertices, but it is often defined as a separate set, with the endvertices of an edge specified by an incidence relation, or as a multiset of two-element subsets. Edges may be endowed with direction, leading to the notion of a dire ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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See List of eponyms (L-Z) An asterisk designates people who became eponyms despite their stated wishes not to. ...

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List of eponyms, List of eponyms - A, List of eponyms - B, List of eponyms - C, List of eponyms - D, List of eponyms - E, List of eponyms - F, List of eponyms - G, List of eponyms - H, List of eponyms - I - J, List of eponyms - K, List of eponyms - L - Z

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Labyrinth is a word of pre-Greek ("Pelasgian") origin absorbed by classical Greek, and is apparently related to labrys, a word for the archaic iconic "double axe", with -inthos connoting "place" (as in "Corinth"). The complex palace of Knossos in Crete is usually implicated, though the actual dancing-ground, depicted in frescoed patterns at Knossos, has not been found. Something was being shown to visitors as a labyrinth at Knossos in the 1st century AD (Philostratos, De vita Apollonii T ...

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Labyrinth, Labyrinth - Ancient labyrinths, Labyrinth - Labyrinth as pattern, Labyrinth - Modern labyrinths, Labyrinth - Modern interpretations of the Greek labyrinth, Labyrinth - Cultural meanings

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A crossing is a pair of intersecting edges. A graph is embeddable on a surface if its vertices and edges can be arranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph can embed. A planar graph is one which can be drawn on the (Euclidean) plane without any crossing; and a plane graph, one which is drawn in such fashion. In other words, a planar graph is a graph of genus 0. The example graph is planar; the complete graph on n vertices, for n> 4, is not p ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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The distance dG(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortest path between them. The subscript G is usually dropped when there is no danger of confusion. When u and v are identical, their distance is 0. When u and v are unreachable from each ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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An arc, or directed edge, is an ordered pair of endvertices. In such ordered pair, the first vertex is called a head, or initial vertex; and the second one, a tail, or terminal vertex. It can be thought of as an edge associated with a direction, namely designating a head and a tail to the endvertices. An undirected edge disregards any sense of direction and treats both endvertices interchangeably. A loop in a digraph, however, keeps a sense of direction and treats both head and tail ide ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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In modern imagery, the labyrinth is often confused with the maze, in which one may become lost. The myth of the labyrinth has in recent times transformed into a stage play by Ilinka Crvenkovska in which exploring notions of a man's ability to control his own fate, Theseus in an act of suicide is killed by the Minotaur only to be killed himself by the horrified towns people. The Argentinian writer Jorge Luis Borges was entranced with the idea of the labyrinth, and used it extensively throughout his short stories. His modern lite ...

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Labyrinth, Labyrinth - Ancient labyrinths, Labyrinth - Labyrinth as pattern, Labyrinth - Modern labyrinths, Labyrinth - Modern interpretations of the Greek labyrinth, Labyrinth - Cultural meanings

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Connectivity extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency. If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected; otherwise, the graph is disconnected. A graph is totally disconnected if there is no path connecting any pair of vertices. This is just an ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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In recent years, there has been a resurgence of interest in the labyrinth symbol, which has inspired a revival in labyrinth building, notably at Willen Park, Milton Keynes; Grace Cathedral, San Francisco; Tapton Park, Chesterfield; and the Labyrinthe de Harbor 16 in Montreal. Countless computer games depict mazes and labyrinths. The internet is a huge labyrinth. ...

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Labyrinth, Labyrinth - Ancient labyrinths, Labyrinth - Labyrinth as pattern, Labyrinth - Modern labyrinths, Labyrinth - Modern interpretations of the Greek labyrinth, Labyrinth - Cultural meanings

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In Antiquity the more complicated labyrinth pattern familiar from medieval examples was already developed. In Roman floor mosaics the simple classical labyrinth is framed in the meander border pattern, squared off as the medium requires, but still recognisable. Often an image of a bull-man, a minotaur, appears in the centre of these mosaic labyrinths. Roman meander patterns gradually developed in complexity towards the four-fold shape that is now familiarly known as the medieval form. The labyrinth retains its connection with death an ...

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Labyrinth, Labyrinth - Ancient labyrinths, Labyrinth - Labyrinth as pattern, Labyrinth - Modern labyrinths, Labyrinth - Modern interpretations of the Greek labyrinth, Labyrinth - Cultural meanings

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In graph theory, degree, especially that of a vertex, is usually a measure of immediate adjacency. An edge connects two vertices; these two vertices are said to be incident to that edge, or, equivalently, that edge incident to those two vertices. All degree-related concepts have to do with adjacency or incidence. The degree, or valency, dG(v) of a vertex v in a graph G is the number of edges incident to v, with loops being counted twice. A vertex of degre ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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