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.

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 graphs (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). 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.

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