A uniquely Hamiltonian graph is a graph possessing a single Hamiltonian cycle.
Classes of uniquely Hamiltonian graphs include the cycle graphs 
,
Hanoi graphs 
, ladder graphs 
(for 
), sun graphs, and uniquely
pancyclic graphs.
The numbers of uniquely Hamiltonian graphs on 
, 2, ... are 0, 0, 1, 2, 3, 12, 49, 482, 6380, ... (OEIS
A307956) and the corresponding numbers of planar
uniquely Hamiltonian graphs are 0, 0, 1, 2, 3, 12, 49, 460, 4994, ... (OEIS A307957).
A Hamiltonian cubic graph contains at least three Hamiltonian cycles, so there are no uniquely Hamiltonian cubic graphs (Goedgebeura et al. 2019).
