We give constructive proofs for the existence of uniquely hamiltonian graphs
for various sets of degrees. We give constructions for all sets with minimum 2
(a
  trivial case), all sets with minimum 3 that contain an even number (for sets
without an even number it is known that no uniquely hamiltonian graphs exist),
and
  all sets with at least two elements and minimum 4 where all other elements
are at least 10. For minimum degree 3 and 4, the constructions also give
3-connected graphs