Hamiltonian path
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In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph
which visits each vertex exactly once. A Hamiltonian cycle is a cycle in an undirected graph
which visits each vertex exactly once and also returns to the starting vertex. Determining
whether such paths and cycles exist in graphs is the Hamiltonian path problem which is NPcomplete.
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.
Contents
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1 Definition
2 Examples
3 Notes
4 Bondy-Chvátal theorem
5 See also
6 External links
7 References
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Definition
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 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 (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 the same direction, i.e., connected tail-to-head.
A "Hamiltonian decomposition" is an edge decomposition of a graph into Hamiltonian circuits.
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Examples
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a complete graph with more than two vertices is Hamiltonian
every circle graph is Hamiltonian
every tournament has an odd number of Hamiltonian paths
every platonic solid, considered as a graph, is Hamiltonian
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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.
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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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 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 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 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 > 2) is Hamiltonian if the sum of the degree of two nonadjacent vertices is n or greater.
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See also
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Eulerian path
Traveling Salesman Problem
Lovász conjecture
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External links
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The Hamiltonian Page - Hamiltonian cycle and path problems,
their generalizations and variations.
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Weisstein, Eric W., Hamiltonian Circuit. Wolfram Research, 2003.
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References
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Ore, O "A Note on Hamiltonian Circuits." Amer. Math. Monthly 67, 55, 1960.
DeLeon, Melissa, "A Study of Sufficient Conditions for Hamiltonian Cycles". Department
of Mathematics and Computer Science, Seton Hall University, South Orange, New
Jersey, United States of America. [PDF]
Hamilton, William Rowan, "Memorandum respecting a new system of roots of unity".
Philosophical Magazine, 12 1856
Hamilton, William Rowan, "Account of the Icosian Calculus". Proceedings of the Royal
Irish Academy, 6 1858
Retrieved from "http://en.wikipedia.org/wiki/Hamiltonian_path"
Categories: Graph theory | NP-complete problems
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