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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-byy0c

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

In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-lmzedk
In[3]:=3
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-finxt2
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-joq4a6
Out[4]=4

This function has been superseded by FindHamiltonianCycle in the Wolfram System:

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-qs9cop
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-s1us5t
Out[2]=2

Options  (2)Common values & functionality for each option

MaxIterations  (1)

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-d5urw

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

In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-yo7g01
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-orgpti
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-karyko

RandomSeed  (1)

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-9243h

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

In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-x204sn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-z026u6
Out[3]=3

Applications  (1)Sample problems that can be solved with this function

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

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-tio52j
In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-6d3gmj
In[3]:=3
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-s6pel
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-hhyghm
Out[4]=4

Properties & Relations  (1)Properties of the function, and connections to other functions

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:

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-qdtybq
In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-zw5xnc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-yjrcq0
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-zxq64f
In[2]:=2
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-od59ay
In[3]:=3
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-g5tm4w
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ye753hcr6n0hpe2iquhq-7aaigl
Out[4]=4
Wolfram Research (2007), FindHamiltonianCycle, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html.
✖
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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_findhamiltoniancycle, author="Wolfram Research", title="{FindHamiltonianCycle}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html}", note=[Accessed: 09-July-2025 ]}

✖
@misc{reference.wolfram_2025_findhamiltoniancycle, author="Wolfram Research", title="{FindHamiltonianCycle}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html}", note=[Accessed: 09-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_findhamiltoniancycle, organization={Wolfram Research}, title={FindHamiltonianCycle}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html}, note=[Accessed: 09-July-2025 ]}

✖
@online{reference.wolfram_2025_findhamiltoniancycle, organization={Wolfram Research}, title={FindHamiltonianCycle}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/FindHamiltonianCycle.html}, note=[Accessed: 09-July-2025 ]}