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

open allclose all

## Basic Examples(2)

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

HamiltonianCycles has been superseded by FindHamiltonianCycle:

## Scope(1)

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

This plots the graph and highlights the cycle in red:

This finds all Hamiltonian cycles:

## Applications(1)

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

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

## Properties & Relations(1)

A graph that has a Hamiltonian cycle must be biconnected:

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

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_2024_hamiltoniancycles, author="Wolfram Research", title="{HamiltonianCycles}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html}", note=[Accessed: 18-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_hamiltoniancycles, organization={Wolfram Research}, title={HamiltonianCycles}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/HamiltonianCycles.html}, note=[Accessed: 18-June-2024 ]}