The numbers of uniquely Hamiltonian graphs on , 2, ... are 0, 0, 1, 2, 3, 12, 49, 482, 6380, ... (OEIS
A307956; Goedgebeur et al. 2019, Table
4) and the corresponding numbers of planar uniquely
Hamiltonian graphs are 0, 0, 1, 2, 3, 12, 49, 460, 4994, ... (OEIS A307957;
Goedgebeur et al. 2019, Table 1).
A Hamiltonian cubic graph contains at least three Hamiltonian cycles, so there are no uniquely Hamiltonian cubic graphs (Goedgebeur
et al. 2019).
Fleischner (2014) 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). In this
work, graphs related to this construction are known as Fleischner
graphs.
Barefoot and Entringer (1981) proved that for every , there exist exactly uniquely hamiltonian graphs of maximum size (Goedgebeur
et al. 2019).
There are no -regular uniquely Hamiltonian graphs for (Haxell et al. 2007, Fleischner 2014, Brinkmann
and De Pauw 2024).
Barefoot, C. A. and Entringer, R. C. "A Census of Maximum Uniquely Hamiltonian Graphs." J. Graph Th.5, 315-321,
1981.Bondy, J. A. and Jackson, B. "Vertices of Small Degree
in Uniquely Hamiltonian Graphs." J. Combin. Th., Ser. B74, 265-275,
1998.Brinkmann, G. and De Pauw, M. "Uniquely Hamiltonian Graphs
for Many Sets of Degrees." Disc. Math. Comput. Sci.26, No. 3,
#7, 2024. https://arxiv.org/abs/2304.08946.Fleischner,
H. "Uniquely Hamiltonian Graphs of Minimum Degree 4." J. Graph Th.75,
167-177, 2014.Goedgebeur, J.; Meersman, B.; and Zamfirescu, C. T.
"Graphs with Few Hamiltonian Cycles." 15 Jul 2019. https://arxiv.org/abs/1812.05650.Haxell,
P.; Seamone, D.; and Verstraete,J. " Independent Dominating Sets and Hamiltonian
Cycles." J. Graph Th.54, 233-234, 2007.House of
Graphs. "Uniquely Hamiltonian Graphs." https://houseofgraphs.org/meta-directory/uniquely-hamiltonian.Sloane,
N. J. A. Sequences A307956 and A307957 in "The On-Line Encyclopedia of Integer
Sequences."
,
Hanoi graphs 
, ladder graphs 
(for 
), sun graphs, and uniquely
pancyclic graphs.
, 2, ... are 0, 0, 1, 2, 3, 12, 49, 482, 6380, ... (OEIS
A307956; Goedgebeur et al. 2019, Table
4) and the corresponding numbers of planar uniquely
Hamiltonian graphs are 0, 0, 1, 2, 3, 12, 49, 460, 4994, ... (OEIS A307957;
Goedgebeur et al. 2019, Table 1).
, there exist exactly ![2^([n/2]-4)](/images/equations/UniquelyHamiltonianGraph/Inline7.svg)
uniquely hamiltonian graphs of maximum size (Goedgebeur
et al. 2019).
-regular uniquely Hamiltonian graphs for 
(Haxell et al. 2007, Fleischner 2014, Brinkmann
and De Pauw 2024).
