Planar Hypohamiltonian Graph


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 countgraphreference(s)
9494-Thomassen graphs
57Hatzel graphHatzel (1979)
4848-Zamfirescu graphZamfirescu and Zamfirescu (2007)
42Wiener-Araya graphWiener and Araya (2009)
42179 graphsJooyandeh et al. (2017)
4025 graphsJooyandeh et al. (2017)
382 graphsTsai (2024)
376 graphsTsai (2024)
342 graphsTsai (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 graphs (including the Wiener-Araya graph), and 496 on 43 vertices. They also showed that there exist planar hypohamiltonian graphs of order n for every n>=42 (Jooyandeh et al. 2017), a result subsequently improved to n>=40 (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.

Every planar hypohamiltonian graph contains a vertex of degree 3 (Thomassen 1978), and letting kappa(G) be the vertex connectivity, delta(G) the minimum vertex degree, and lambda(G) the edge connectivity of a planar hypohamiltonian graph G,


(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).

See also

Cubic Planar Hypohamiltonian Graph, Hatzel Graph, Hypohamiltonian Graph, Planar Graph, Planar Hypotraceable Graph, Thomassen Graphs, Wiener-Araya Graph, Zamfirescu Graphs

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Cite this as:

Weisstein, Eric W. "Planar Hypohamiltonian Graph." From MathWorld--A Wolfram Web Resource.

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