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A Hamiltonian cycle in a dodecahedron. Like all platonic solids, the dodecahedron is Hamiltonian.
A Hamiltonian cycle in a dodecahedron. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal Like all platonic solids, the dodecahedron is Hamiltonian. In Geometry, a Platonic solid is a convex Regular polyhedron.
A Hamiltonian path (black) over a graph (blue).
A Hamiltonian path (black) over a graph (blue).

In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph which visits each vertex exactly once. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Graph theory, a path in a graph is a Sequence of vertices such that from each of its vertices there is an edge to the next vertex In Mathematics and Computer science, a graph is the basic object of study in Graph theory. For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out A Hamiltonian cycle (or Hamiltonian circuit) is a cycle in an undirected graph which visits each vertex exactly once and also returns to the starting vertex. Cycle in Graph theory and Computer science has several meanings A closed walk with repeated vertices allowed In Mathematics and Computer science, a graph is the basic object of study in Graph theory. For other uses see Vertex. In Graph theory, a vertex (plural vertices) or node is the fundamental unit out Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem which is NP-complete. In the mathematical field of Graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian In Computational complexity theory, the Complexity class NP-complete (abbreviated NP-C or NPC) is a class of problems having two properties

Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the Icosian Game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal Hamilton solved this problem using the Icosian Calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). The Icosian Calculus is a non-commutative Algebraic structure discovered by the Irish mathematician William Rowan Hamilton in 1856. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Unfortunately, this solution does not generalize to arbitrary graphs.

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Definitions

A Hamiltonian path or traceable path is a path that visits each vertex exactly once. In Graph theory, a path in a graph is a Sequence of vertices such that from each of its vertices there is an edge to the next vertex A graph that contains a Hamiltonian path is called a traceable graph. A graph is Hamilton-connected if for every pair of vertices there is a Hamiltonian path between the two vertices.

A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (except the vertex which is both the start and end, and so is visited twice). Cycle in Graph theory and Computer science has several meanings A closed walk with repeated vertices allowed A graph that contains a Hamiltonian cycle is called a Hamiltonian graph.

Similar notions may be defined for directed graphs, where each edge (arc) of a path or cycle can only be traced in a single direction (i. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. e. , the vertices are connected with arrows and the edges traced "tail-to-head").

A Hamiltonian decomposition is an edge decomposition of a graph into Hamiltonian circuits. Graph theory is a growing area in mathematical research and has a large specialized vocabulary

Examples

Notes

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. In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an In Graph theory, a cycle graph is a graph that consists of a single cycle, or in other words some number of vertices connected in a closed chain A tournament is a Directed graph obtained by choosing a direction for each edge in an Undirected Complete graph. In Geometry, a Platonic solid is a convex Regular polyhedron.

The line graph of a Hamiltonian graph is Hamiltonian. In a Graph theory, the line graph L ( G) of an Undirected graph G is another graph L ( G) that represents the The line graph of an Eulerian graph is Hamiltonian. In Graph theory, an Eulerian path is a path in a graph which visits each edge exactly once

A tournament (with more than 2 vertices) is Hamiltonian if and only if it is strongly connected. A Directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex

A Hamiltonian cycle may be used as the basis of a zero-knowledge proof. In Cryptography, a zero-knowledge proof or zero-knowledge protocol is an interactive method for one party to prove to another that a (usually mathematical statement

Bondy-Chvátal theorem

The best characterization of Hamiltonian graphs was given in 1972 by the Bondy-Chvátal theorem which generalizes earlier results by G. A. Dirac and Øystein Ore. Václav (Vašek Chvátal (born 1946 (ˈvaːt͡slaf ˈxvaːtal is a professor in the Department of Computer Science and Software Engineering at Concordia University in Gabriel Andrew Dirac (1925–1984 was a Mathematician who mainly worked in Graph theory. Øystein Ore ( 7 October 1899 in Oslo, Norway &ndash 13 August 1968 in Oslo was a Norwegian Mathematician 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. In Mathematics and Computer science, a graph is the basic object of study in Graph theory.

Bondy-Chvátal theorem (1972)

A graph is Hamiltonian if and only if its closure is Hamiltonian.

As complete graphs are Hamiltonian, all graphs whose closure is complete are Hamiltonian, which is the content of the following earlier theorems by Dirac and Ore.

Dirac (1952)

A simple graph with n vertices (n > 2) is Hamiltonian if each vertex has degree n/2 or greater.

Ore (1960)

A graph with n vertices (n > 3) is Hamiltonian if, for each pair of non-adjacent vertices, the sum of their degrees is n or greater (see Ore's theorem). Ore's Theorem is a result in Graph theory due to Øystein Ore (1960

The following theorems can be regarded as directed versions:

Ghouila-Houiri (1960)

A strongly connected simple directed graph with n vertices is Hamiltonian or some vertex has a full degree smaller than n. A Directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex

Meyniel (1973)

A strongly connected simple directed graph with n vertices is Hamiltonian or the sum of full degrees of some two distinct non-adjacent vertices is smaller than 2n − 1. A Directed graph is called strongly connected if there is a path from each vertex in the graph to every other vertex

It should be noted that the number of vertices must be doubled because each undirected edge corresponds to two directed arcs and thus the degree of a vertex in the directed graph is twice the degree in the undirected graph.

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Dictionary

Hamiltonian path

-noun

  1. (graph theory) A path through an undirected graph which visits each vertex exactly once.
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