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HamiltonianCycles[g,n]

gives a list of n Hamiltonian cycles.

HamiltonianCycles[g]

gives a list of one Hamiltonian cycle.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
See Also
Tech Notes
Related Guides
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GraphUtilities`
GraphUtilities`

HamiltonianCycles

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

HamiltonianCycles[g,n]

gives a list of n Hamiltonian cycles.

HamiltonianCycles[g]

gives a list of one Hamiltonian cycle.

Details and Options

  • HamiltonianCycles functionality is now available in the built-in Wolfram Language function FindHamiltonianCycle.
  • To use HamiltonianCycles, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
  • HamiltonianCycles[g,n] returns an empty list if no Hamiltonian cycle exists.
  • HamiltonianCycles considers the input graph as undirected.
  • The complexity of the algorithm is such that finding all Hamiltonian cycles for a large graph can take an exponential amount of time.

Examples

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Basic Examples  (2)

Wolfram Language code: Needs["GraphUtilities`"]

This defines a small graph and finds a Hamiltonian cycle of the graph:

Wolfram Language code: g = {1 -> 3, 1 -> 2, 1 -> 4, 2 -> 3, 2 -> 4};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]
Wolfram Language code: HamiltonianCycles[g]

HamiltonianCycles has been superseded by FindHamiltonianCycle:

Wolfram Language code: g = UndirectedGraph[Graph[{1 -> 3, 1 -> 2, 1 -> 4, 2 -> 3, 2 -> 4}]]
Wolfram Language code: FindHamiltonianCycle[g][[All, All, 1]]

Scope  (1)

Wolfram Language code: Needs["GraphUtilities`"]

This defines a small graph and finds a Hamiltonian cycle of the graph:

Wolfram Language code: g = {0 -> 1, 2 -> 3, 2 -> 1, 3 -> 0, 4 -> 5, 4 -> 7, 4 -> 3, 5 -> 6, 5 -> 2, 6 -> 7, 6 -> 1, 7 -> 0};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]
Wolfram Language code: c = HamiltonianCycles[g]

This plots the graph and highlights the cycle in red:

Wolfram Language code: cs = Transpose[{c[[1]], RotateRight[c[[1]]]}];GraphPlot[g, EdgeRenderingFunction -> (If[MemberQ[cs, #2] || MemberQ[cs, Reverse[#2]], {Red, Thickness[.01], Line[#1]}, {Black, Line[#1]}]&), VertexLabeling -> True]

This finds all Hamiltonian cycles:

Wolfram Language code: HamiltonianCycles[g, All]

Applications  (1)

This finds all possible Hamiltonian cycles in the graph consisting of bordering countries in South America:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: g = Flatten[Map[(If[MemberQ[CountryData["SouthAmerica"], #[[1]]] && MemberQ[CountryData["SouthAmerica"], #[[2]]], #, {}])&, Flatten[Thread[# -> CountryData[#, "BorderingCountries"]]& /@ CountryData["SouthAmerica"]]]];
Wolfram Language code: c = HamiltonianCycles[g, All]

This shows the first of these two cycles; the second is just a reversal of the first:

Wolfram Language code: cs = Transpose[{c[[2]], RotateRight[c[[2]]]}];GraphPlot[g, EdgeRenderingFunction -> (If[MemberQ[cs, #2] || MemberQ[cs, Reverse[#2]], {Red, Thickness[.01], Line[#1]}, {Black, Line[#1]}]&), VertexLabeling -> True, MultiedgeStyle -> False]

Properties & Relations  (1)

A graph that has a Hamiltonian cycle must be biconnected:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: g = {1 -> 2, 2 -> 3, 3 -> 4, 4 -> 1, 1 -> 5};
Wolfram Language code: Bicomponents[g]
Wolfram Language code: HamiltonianCycles[g]

A graph that is biconnected does not necessarily have a Hamiltonian cycle:

Wolfram Language code: g = {1 -> 2, 1 -> 4, 1 -> 5, 2 -> 3, 3 -> 6, 4 -> 6, 5 -> 6};
Wolfram Language code: Bicomponents[g]
Wolfram Language code: HamiltonianCycles[g]
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]
Wolfram Research (2007), HamiltonianCycles, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html.

Text

Wolfram Research (2007), HamiltonianCycles, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html.

CMS

Wolfram Language. 2007. "HamiltonianCycles." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html.

APA

Wolfram Language. (2007). HamiltonianCycles. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html

BibTeX

@misc{reference.wolfram_2026_hamiltoniancycles, author="Wolfram Research", title="{HamiltonianCycles}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_hamiltoniancycles, organization={Wolfram Research}, title={HamiltonianCycles}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html}, note=[Accessed: 15-June-2026]}