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FindHamiltonianCycle[g]

attempts to find a Hamiltonian cycle.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Options  
MaxIterations  
RandomSeed  
Applications  
Properties & Relations  
Possible Issues  
See Also
Tech Notes
Related Guides
Cite this Page
GraphUtilities`
GraphUtilities`

FindHamiltonianCycle

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

FindHamiltonianCycle[g]

attempts to find a Hamiltonian cycle.

Details and Options

  • FindHamiltonianCycle functionality is now available in the built-in Wolfram Language function FindHamiltonianCycle.
  • To use FindHamiltonianCycle, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
  • FindHamiltonianCycle[g] returns an empty list if no Hamiltonian cycle is found.
  • FindHamiltonianCycle considers the input graph as undirected.
  • FindHamiltonianCycle uses heuristic algorithms to find a Hamiltonian cycle; therefore, there is no guarantee that a Hamiltonian cycle will be found even if one exists.

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: FindHamiltonianCycle[g]

This function has been superseded by FindHamiltonianCycle in the Wolfram System:

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

Options  (2)

MaxIterations  (1)

Wolfram Language code: Needs["Combinatorica`"]

This limits the maximum number of iterations to try to find a Hamiltonian cycle:

Wolfram Language code: FindHamiltonianCycle[GridGraph[8, 8], MaxIterations -> 5]//Timing
Wolfram Language code: FindHamiltonianCycle[GridGraph[8, 8]]//Timing
Wolfram Language code: FindHamiltonianCycle[GridGraph[8, 8], MaxIterations -> 5]//Timing

RandomSeed  (1)

Wolfram Language code: Needs["Combinatorica`"]

This specifies a random seed different from the default to try to find a Hamiltonian cycle:

Wolfram Language code: FindHamiltonianCycle[GridGraph[10, 10], RandomSeed -> 123]
Wolfram Language code: FindHamiltonianCycle[GridGraph[10, 10]]

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 = FindHamiltonianCycle[g]

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

Wolfram Language code: cs = Transpose[{c, RotateRight[c]}];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)

FindHamiltonianCycle uses heuristic algorithms. Unlike HamiltonianCycles, it is not guaranteed to find a Hamiltonian cycle even when one exists. But for large graphs, FindHamiltonianCycle can sometimes be faster at finding one cycle.

This defines a graph of 500 vertices, and uses these two functions to find a Hamiltonian cycle:

Wolfram Language code: SeedRandom[123]; g = RandomGraph[500, 0.3];
Wolfram Language code: FindHamiltonianCycle[g]//Short//Timing
Wolfram Language code: HamiltonianCycles[g]//Short//Timing

Possible Issues  (1)

FindHamiltonianCycle uses heuristic algorithms, which are not guaranteed to find a Hamiltonian cycle even when one exists:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: Needs["Combinatorica`"]
Wolfram Language code: FindHamiltonianCycle[GridGraph[10, 10]]
Wolfram Language code: HamiltonianCycles[GridGraph[10, 10]]
Wolfram Research (2007), FindHamiltonianCycle, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_findhamiltoniancycle, author="Wolfram Research", title="{FindHamiltonianCycle}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_findhamiltoniancycle, organization={Wolfram Research}, title={FindHamiltonianCycle}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html}, note=[Accessed: 15-June-2026]}