A planar hypohamiltonian graph is a hypohamiltonian graph that is also planar. A number of planar hypohamiltonian graphs are illustrated above.
Chvátal (1973) first asked if there existed planar hypohamiltonian graphs, and in fact it was conjectured that such graphs might not exist (Grünbaum 1974, Jooyandeh et al. 2017). An infinite family was subsequently found by Thomassen (1976), the smallest among them having 105 vertices. The following table summarizes incrementally smallest known planar hypohamiltonian graphs.
| vertex count | graph | reference(s) |
| 94 | 94-Thomassen graph | |
| 57 | Hatzel graph | Hatzel (1979) |
| 48 | 48-Zamfirescu graph | Zamfirescu and Zamfirescu (2007) |
| 42 | Wiener-Araya graph | Wiener and Araya (2009) |
| 42 | 179 graphs | Jooyandeh et al. (2017) |
| 40 | 25 graphs | Jooyandeh et al. (2017) |
| 38 | 2 graphs | Tsai (2024) |
| 37 | 6 graphs | Tsai (2024) |
| 34 | 2 graphs | Tsai (2024) |
Jooyandeh et al. (2017) searched for "planar 4-face deflatable hypohamiltonian graphs" and found 25 on 40 vertices (all
of girth 4), 179 on 42 vertices
(including the Wiener-Araya graph), and 496
on 43 vertices. They also showed that there exist
planar hypohamiltonian graphs of order 
for every 
(Jooyandeh et al. 2017), a result subsequently
improved to 
(Tsai 2024) and 
(Durias et al. 2026). It is not known if 34
is the smallest possible vertex count of a planar
hypohamiltonian, with 23 being the best currently known lower
bound (Goedgebeur and Zamfirescu 2017; improving previous bounds from Aldred
et al. 1997, Jooyandeh et al. 2017).
There are no hypohamiltonian planar graphs with girth 5 on fewer than 45 vertices, and there is exactly one such graph on 45 vertices (Jooyandeh et al. 2017), illustrated above.
Thomassen (1976, 1978) proved that every planar hypohamiltonian graph contains a vertex of degree 3, and Zamfirescu (2019) proved that every planar hypohamiltonian graph contains at least four cubic vertices (Tsai 2024).
Letting 
be the vertex connectivity, 
the minimum vertex
degree, and 
the edge connectivity
of a planar hypohamiltonian graph 
,
 |
(Jooyandeh et al. 2017). Furthermore, a planar hypohamiltonian graph can have girth at most 5 (Jooyandeh et al. 2017).
A number of cubic planar hypohamiltonian graphs have been found by Araya and Wiener (2011), McKay and Jooyandeh (McKay), and Tsai (2024).
Durias et al. (2026) also used local modifications to obtain planar almost hypohamiltonian graphs of every order 
and planar
hypotraceable graphs of every order 
. The authors devised the main approach and generated
the examples by hand. AI tools supplied code for checking Hamiltonian
cycles and searching with plantri.
