A graph
is hypohamiltonian if is nonhamiltonian,
but
is Hamiltonian for every (Bondy and Murty 1976, p. 61). The Petersen
graph, which has ten nodes, is the smallest hypohamiltonian graph (Herz et
al. 1967; Bondy and Murty 1976, p. 61) and the only such graph on ten nodes.
Herz et al. (1967) showed that there are no hypohamiltonian graphs with 11
or 12 vertices.
By parity, no bipartite graph is hypohamiltonian.
Many (but not all) snarks are hypohamiltonian.
Zamfirescu (2019) showed that every singlecross
hypohamiltonian graph contains a cubic vertex (Tsai 2024).
The Lindgren-Sousselier graphs are a sequence of hypohamiltonian graphs on vertices constructed independently by Sousselier (in Herz
et al. 1967) and Lindgren (1967).
Bondy (1972) found an infinite sequence of hypohamiltonian graphs on vertices.
Sousselier found a cubic hypohamiltonian graph on 18 vertices (Chvátal 1973). Chvátal (1973) showed there exists a hypohamiltonian graph on vertices for every . More generally, there exists a hypohamiltonian graph
for every
with the exception of (Collier and Schmeichel 1978) and (Aldred et al. 1997). Aldred et al. (1997)
give a complete enumeration of all (seven) hypohamiltonian graphs on 17 or fewer
vertices. McKay gives a listing of all known hypohamiltonian graphs up to 26 vertices
(where the enumerations on 18 vertices and higher may be incomplete), as well as
a girth-restricted subset of cubic hypohamiltonian up to 26 vertices. The numbers
of hypohamiltonian graphs on , 2, ... nodes are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1,
0, 1, 4, 0, 14, 34, ... (OEIS A141150; Goedgebeur
and Zamfirescu 2017).
Thomassen (1973) found hypohamiltonian graphs on and 25 vertices, and Collier and Schmeichel (1978) found
"new" hypohamiltonian graphs for and 22, though their "new" graph is actually isomorphic to the first Blanuša
snark and their "new" graphs are isomorphic to the Loupekine
snarks.
A number of non-snark hypohamiltonian graphs, including
a number previously discussed, are shown above, ordered by vertex count.
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SIAM J. Disc. Math.13, 25-32, 2000.Aldred, R. E. L.;
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33-41, 1973.Collier, J. B. and Schmeichel, E. F. "Systematic
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F. Graph
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F. "Recherche systématique des graphes hypohamiltoniens." In Theory
of Graphs: Internat. Sympos., Rome 1966 (Ed. P. Rosenstiehl). Paris: Gordon
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Petersen Graph. Cambridge, England: Cambridge University Press, 1993.House
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Hypohamiltonian Graphs." J. Graph Th.90, 189-207, 2019.
is hypohamiltonian if 
is nonhamiltonian,
but 
is Hamiltonian for every 
(Bondy and Murty 1976, p. 61). The Petersen
graph, which has ten nodes, is the smallest hypohamiltonian graph (Herz et
al. 1967; Bondy and Murty 1976, p. 61) and the only such graph on ten nodes.
Herz et al. (1967) showed that there are no hypohamiltonian graphs with 11
or 12 vertices.
vertices constructed independently by Sousselier (in Herz
et al. 1967) and Lindgren (1967).
vertices.
vertices for every 
. More generally, there exists a hypohamiltonian graph
for every 
with the exception of 
(Collier and Schmeichel 1978) and 
(Aldred et al. 1997). Aldred et al. (1997)
give a complete enumeration of all (seven) hypohamiltonian graphs on 17 or fewer
vertices. McKay gives a listing of all known hypohamiltonian graphs up to 26 vertices
(where the enumerations on 18 vertices and higher may be incomplete), as well as
a girth-restricted subset of cubic hypohamiltonian up to 26 vertices. The numbers
of hypohamiltonian graphs on 
, 2, ... nodes are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1,
0, 1, 4, 0, 14, 34, ... (OEIS A141150; Goedgebeur
and Zamfirescu 2017).
and 25 vertices, and Collier and Schmeichel (1978) found
"new" hypohamiltonian graphs for 
and 22, though their "new" 
graph is actually isomorphic to the first Blanuša
snark and their "new" 
graphs are isomorphic to the Loupekine
snarks.
