HamiltonianCycles

gives a list of n Hamiltonian cycles.

HamiltonianCycles[g]
gives a list of one Hamiltonian cycle.

The complexity of the algorithm is such that finding all Hamiltonian cycles for a large graph can take an exponential amount of time.

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:

Needs["GraphUtilities`"]

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

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This plots the graph and highlights the cycle in red:

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This finds all Hamiltonian cycles:

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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:

A graph that has a Hamiltonian cycle must be biconnected:

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