  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 added for completeness), 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 minimum 4 and at least two
elements, where all degrees different from 4 are at least 10. For minimum
degree 3 and 4, the constructions also give 3-connected graphs. We also
introduce the concept of seeds, which makes the above results possible and
might be useful in the study of Sheehan's conjecture. Furthermore we prove that
3-connected uniquely hamiltonian 4-regular graphs exist if and only if
2-connected uniquely hamiltonian 4-regular graphs exist.
