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In graph theory, a Hamiltonian path (also called traceable path) is a path that visits each vertex exactly once.

A Hamiltonian cycle (also called Hamiltonian circuit, vertex tour or graph cycle) is a cycle that visits each vertex exactly once, except for the starting vertex.

A graph that contains a Hamiltonian path is called a traceable graph. A graph that contains a Hamiltonian cycle is called a Hamiltonian graph. 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.

Similar notions may be defined for directed graphs, where edges (arcs) of a path or a cycle are required to point in the same direction, i.e., connected tail-to-head.

The Hamiltonian cycle problem or Hamiltonian circuit problem in graph theory is to find a Hamiltonian cycle in a given graph. The Hamiltonian path problem is to find a Hamiltonian path in a given graph.

There is a simple relation between the two problems. The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G.

Both problems are NP-complete. However, certain classes of graphs always contain Hamiltonian paths. For example, it is known that every tournament has an odd number of Hamiltonian paths.

The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to unity if they are adjacent and infinity otherwise.

1 History

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. 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). Unfortunately, this solution does not generalize to arbitrary graphs.

2 See also

3 External links and resources

4 References


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Topics: Hamiltonian Path, History, See Also, External Links And...

Hamiltonian (quantum mechanics)The Hamiltonian denoted H has two distinct but closely related meanings. In classical mechanics, it is a function that describes the state of a mechanical system in terms of position and momentum variables (i. symplectic variables), which is the basis forHamiltonian groupIn group theory, a non-abelian group G is called Hamiltonian if every subgroup of G is normal. Clearly, every abelian group has this property, because all subgroups of an abelian group are normal subgroups, but there are non-abelian examples as well.Hamiltonian mechanicsHamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. It arose from Lagrangian mechanics, another re-formulation of classical mechanics, introduced by Joseph Louis Lagrange in 1788. It can ho
Hamiltonian pathIn graph theory, a Hamiltonian path (also called traceable path is a path that visits each vertex exactly once. A Hamiltonian cycle (also called Hamiltonian circuit vertex tour or graph cycle is a cycle that visits each vertex exactly once, except for theHamiltonianIn physics, Hamiltonian has two distinct but closely related meanings. In classical mechanics, the Hamiltonian is a function describing the state of a mechanical system in terms of position and momentum variables. See Hamiltonian mechanics. In quantum mecHamiltonian constraintIn theories with reparametrization invariance with respect to time, if a canonical Hamiltonian formulation using first class constraints of it is possible (in general, we might have second class constraints), we'd have "gauge" "symmetries" (which really m
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