Fleischner (2014) considered uniquely hamiltonian graphsUniquely Hamiltonian Graph of minimum vertex degree 4. In particular, he constructed uniquely hamiltonian graphs in which every vertex has degree 4 or 14. His smallest examples with connectivity 2 and 3 have 338 and 408 vertices, respectively (Fleischner 2014, Goedgebeur et al. 2019).
The 338-vertex graph begins with a graph 
on 15 vertices and 24 edges that has two Hamiltonian cycles, then builds up a series of graphs 
, 
, and 
in which each 
has 
vertices, 
edges, and two Hamiltonian cycles. He then removes degree-3 edges to produce 
and 
and applies another procedure to remove 
(Knuth 2025, p. 17 and Exercise 120). The culmination of this process is a graph 
on 338 vertices having a unique hamiltonian cycle.
Fleischner also constructed a 30-vertex uniquely Hamiltonian graph used in the proof that there are infinitely many 3-connected uniquely hamiltonian graphs of minimum degree 4 (Fleischner 2014).
Some of the graphs discussed above will be implemented in a future version of the Wolfram Language as GraphData["FleischnerGraphXXX"] for XXX values of 15, 30, 57, 169, and 338.
