The term "Fleischner graphs" is used in this work for graphs arising in the construction by Fleischner (2014) 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 graph with 338 vertices 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 vertices 
and 
of vertex degree 3. 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 uniquely Hamiltonian graph on 30 vertices used in the proof
that infinitely many k-connected graphs
with 
are uniquely Hamiltonian and have minimum vertex degree 4 (Fleischner 2014).
The Fleischner graphs on 15 and 30 vertices have graph crossing number 2 and 4, respectively.
Some of the graphs discussed above are implemented in the Wolfram Language as GraphData["FleischnerGraph15"], GraphData["FleischnerGraph30"], GraphData["FleischnerGraph57"], GraphData["FleischnerGraph169"], and GraphData["FleischnerGraph338"].
