# The + / - Of 144 Notes Of Separation In The Ancient Scales: 12 - 144, 24 - 288, 36 - 432

Discussion in '~THE NEW EXCHANGE~' started by Susan Lynne Schwenger, Sep 28, 2014.

1. ### Susan Lynne SchwengerThe Final Synthesis - isbn 978-0-9939480-0-8Staff Member

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tite graph is hypohamiltonian.

Sousselier (in Herz et al. 1967) and Lindgren (1967) independently constructed the same sequence of hypohamiltonian graphs with 6k+10 vertices, illustrated above (which includes the Petersen graph on 10 vertices).

Bondy (1972) found an infinite sequence of hypohamiltonian graphs on 12k+10 vertices.

Sousselier found a cubic hypohamiltonian graph on 18 vertices (Chvátal 1973). Chvátal (1973) showed there exists a hypohamiltonian graph on p vertices for every p>=26. More generally, there exists a hypohamiltonian graph for every p>=13 with the exception of p=14 (Collier and Schmeichel 1978) and p=17 (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 n=1, 2, ... nodes are 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 4, 0, ... (OEIS A141150), where the next term is thought to be 13 but this has not been proved.

http://mathworld.wolfram.com/HypohamiltonianGraph.html

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2. ### Susan Lynne SchwengerThe Final Synthesis - isbn 978-0-9939480-0-8Staff Member

Messages:
6,656
Chvátal (1973) asked if there existed planar hypohamiltonian graphs.

An infinite family was subsequently found by Thomassen (1976),
the smallest among them having 105 vertices.

Thomassen improved this to 94 vertices,
Hatzel (1979) improved this lower bound to 57 vertices (the Hatzel graph).

Zamfirescu and Zamfirescu (2007) found a planar hypohamiltonian graph
on 48 vertices (the Zamfirescu graph),

and Wiener and Araya (2009) subsequently
found the currently smallest known example
(the Wiener-Araya graph, on 42 vertices).

A number of small planar hypohamiltonian graphs
are illustrated above.

Thomassen (1978) showed that every planar hypohamiltonian graph
contains a vertex of degree 3.

3. ### Susan Lynne SchwengerThe Final Synthesis - isbn 978-0-9939480-0-8Staff Member

Messages:
6,656

Thomassen (1973) found hypohamiltonian graphs on p=20 and 25 vertices,
and Collier and Schmeichel (1978) found "new" hypohamiltonian graphs
for p=18 and 22, though their "new" p=18 graph
is actually isomorphic to the first Blanuša snark
and their "new" p=22 graphs are isomorphic to the Loupekine snarks.

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