Hamiltonian path: Difference between revisions

Browse history interactively

← Previous edit Next edit →

Content deleted Content added

VisualWikitext

Inline

Revision as of 20:48, 12 September 2011

A Hamiltonian path (black) over a graph (blue).

In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle in an undirected graph that 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 NP-complete.

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). This solution does not generalize to arbitrary graphs.

Definitions

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 Hamiltonian-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 that is both the start and end, and so is visited twice). 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.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. This is one of the known but unsolved problems.^{[clarification needed]}

Examples

a complete graph with more than two vertices is Hamiltonian
every cycle graph is Hamiltonian
every tournament has an odd number of Hamiltonian paths
every platonic solid, considered as a graph, is Hamiltonian
every prism is Hamiltonian
The Deltoidal hexecontahedron is the only non-hamiltonian Archimedean dual
Every antiprism is Hamiltonian

Properties

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^{[clarification needed]}.

A tournament (with more than 2 vertices) is Hamiltonian if and only if it is strongly connected.

A Hamiltonian cycle may be used as the basis of a zero-knowledge proof.

Number of different Hamiltonian cycles for a complete graph = (n-1)! / 2.

Number of different Hamiltonian cycles for a complete directed graph = (n-1)!.

Bondy–Chvátal theorem

The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the Bondy–Chvátal theorem, which generalizes earlier results by G. A. Dirac (1952) and Øystein Ore. In fact, both Dirac's and Ore's theorems are less powerful than what can be derived from Pósa's theorem (1962). Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has enough edges. 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 successively adding for all nonadjacent pairs of vertices u and v with degree(v) + degree(u) ≥ n^{[clarification needed]} the new edge uv.

Bondy–Chvátal theorem

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 ≥ 3) is Hamiltonian if each vertex has degree n/2 or greater.^[1]

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).

The following theorems can be regarded as directed versions:

Ghouila-Houiri (1960)

A strongly connected simple directed graph with

n

vertices is Hamiltonian if some vertex has a full degree smaller than

n

.

Meyniel (1973)

A strongly connected simple directed graph with

n

vertices is Hamiltonian if the sum of full degrees of some two distinct non-adjacent vertices is smaller than

2 n - 1

.

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.

Notes

^ Graham, p. 20.

References

Berge, Claude; Ghouila-Houiri, A. (1962), Programming, games and transportation networks, New York: John Wiley & Sons, Inc.
DeLeon, Melissa, "A Study of Sufficient Conditions for Hamiltonian Cycles". Department of Mathematics and Computer Science, Seton Hall University
Dirac, G. A. (1952), "Some theorems on abstract graphs", Proceedings of the London Mathematical Society, 3rd Ser., 2: 69–81, doi:10.1112/plms/s3-2.1.69, MR 0047308
Graham, Ronald L., Handbook of Combinatorics, MIT Press, 1995. ISBN 9780262571708.
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
Meyniel, M. (1973), "Une condition suffisante d'existence d'un circuit hamiltonien dans un graphe orienté", Journal of Combinatorial Theory, Ser. B, 14: 137–147, doi:10.1016/0095-8956(73)90057-9
Ore, O "A Note on Hamiltonian Circuits." American Mathematical Monthly 67, 55, 1960.
Peterson, Ivars, "The Mathematical Tourist". 1988. W. H. Freeman and Company, NY
Pósa, L. A theorem concerning hamilton lines. Magyar Tud. Akad. Mat. Kutato Int. Kozl. 7(1962), 225–226.
Chuzo Iwamoto and Godfried Toussaint, "Finding Hamiltonian circuits in arrangements of Jordan curves is NP-complete," Information Processing Letters, Vol. 52, 1994, pp. 183–189.

External links

[1] Graham, p. 20.

[1]

@@ Line 5: / Line 5: @@
 [[Image:Herschel graph.svg|thumb|The [[Herschel graph]] is the smallest possible [[polyhedral graph]] that does not have a Hamiltonian cycle.]]
-In the [[mathematics|mathematical]] field of [[graph theory]], a '''Hamiltonian path''' (or '''traceable path''') is a [[path (graph theory)|path]] in an [[undirected graph]] that visits each [[vertex (graph theory)|vertex]] exactly once. A '''Hamiltonian cycle''' (or '''Hamiltonian circuit''') is a [[cycle (graph theory)|cycle]] in an [[undirected graph]] which visits each [[vertex (graph theory)|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 [[NP-complete problem|NP-complete]].
+In the [[mathematics|mathematical]] field of [[graph theory]], a '''Hamiltonian path''' (or '''traceable path''') is a [[path (graph theory)|path]] in an [[undirected graph]] that visits each [[vertex (graph theory)|vertex]] exactly once. A '''Hamiltonian cycle''' (or '''Hamiltonian circuit''') is a [[cycle (graph theory)|cycle]] in an [[undirected graph]] that visits each [[vertex (graph theory)|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 [[NP-complete problem|NP-complete]].
 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 [[root of unity|roots of unity]] with many similarities to the [[quaternion]]s (also invented by Hamilton).  This solution does not generalize to arbitrary graphs.
@@ Line 13: / Line 13: @@
 A ''Hamiltonian cycle'', ''Hamiltonian circuit'', ''vertex tour'' or ''graph cycle'' is a [[cycle (graph theory)|cycle]] that visits each vertex exactly once
-(except the vertex which is both the start and end, and so is visited twice). A graph that contains a Hamiltonian cycle is called a '''Hamiltonian graph'''.
+(except the vertex that is both the start and end, and so is visited twice). A graph that contains a Hamiltonian cycle is called a '''Hamiltonian graph'''.
 Similar notions may be defined for ''[[Graph (mathematics)|directed graph]]s'', where each edge (arc) of a path or cycle can only be traced in a single direction (i.e., the vertices are connected with arrows and the edges traced "tail-to-head").
@@ Line 43: / Line 43: @@
 == Bondy–Chvátal theorem ==
-The best vertex degree characterization of Hamiltonian graphs was given in 1972 by the [[J. A. Bondy|Bondy]]–[[Chvátal]] theorem which generalizes earlier results by [[Gabriel Andrew Dirac|G. A. Dirac]] (1952) and [[Øystein Ore]]. In fact, both Dirac's and Ore's theorems are less powerful than what can be derived from [[Lajos Pósa (mathematician)|Pósa]]'s theorem (1962). Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has ''enough edges''. First we have to define the closure of a graph.
+The best vertex degree characterization of Hamiltonian graphs was provided in 1972 by the [[J. A. Bondy|Bondy]]–[[Chvátal]] theorem, which generalizes earlier results by [[Gabriel Andrew Dirac|G. A. Dirac]] (1952) and [[Øystein Ore]]. In fact, both Dirac's and Ore's theorems are less powerful than what can be derived from [[Lajos Pósa (mathematician)|Pósa]]'s theorem (1962). Dirac and Ore's theorems basically state that a graph is Hamiltonian if it has ''enough edges''. 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 successively adding for all [[nonadjacent]] pairs of vertices ''u'' and ''v'' with degree(''v'')&nbsp;+&nbsp;degree(''u'') ≥&nbsp;''n''{{Clarify|date=February 2011}} the new edge ''uv''.