Here is the abstract of a talk that Herbert Fleischner gave at *CombinaTexas: Combinatorics in the South-Central U.S.*, available at [this link][1], in 2003.
The title of his talk was, "*Uniquely Hamiltonian Graphs*":

> <b>Abstract</b>: In the 1970s, J.Sheehan asked whether there are 4-regular simple graphs having precisely one Hamiltonian cycle (= uniquely Hamiltonian graphs). In the early 1990s, 4-regular loopless uniquely Hamiltonian multigraphs were constructed whose underlying uniquely Hamiltonian (simple) graphs have 3- and 4-valent vertices only. C.Thomassen showed that regular graphs of sufficiently high degree cannot be uniquely Hamiltonian; and J.A.Bondy, in his article for the *Handbook of Combinatorics*, asked whether uniquely Hamiltonian graphs must have a vertex of degree 2 or 3. Starting from the examples quoted above, one can construct Eulerian uniquely Hamiltonian graphs of minimum degree 4. 

See also <b>The Open Problems Garden</b>, "[Uniquely Hamiltonian graphs][2]," which highlights the
conjecture:

> If $G$ is a finite $r$-regular graph, where $r > 2$, then $G$ is *not* uniquely Hamiltonian. 


  [1]: http://www.math.tamu.edu/~cyan/combinatexas/2003/abstract.html
  [2]: http://garden.irmacs.sfu.ca/?q=op/uniquely_hamiltonian_graphs