The Fleischner graphs arise in Fleischner's (2014) construction of uniquely Hamiltonian graphs of minimum vertex degree 4. In particular, he constructed uniquely Hamiltonian graphs in which every graph vertex has vertex degree 4 or 14. His smallest examples with vertex 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 defines 
and 
by removing the degree-3 vertices 
and 
. A further construction removes 
and 
in 
(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 infinitely many 3-connected graphs are uniquely Hamiltonian and have minimum vertex degree 4 (Fleischner 2014).
Some of the graphs discussed above are implemented in the Wolfram Language as GraphData["FleischnerGraph15"], GraphData["FleischnerGraph30"], GraphData["FleischnerGraph57"], GraphData["FleischnerGraph169"], and GraphData["FleischnerGraph338"].
