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Number of connected nonhamiltonian graphs with n nodes.
4

%I #29 Jul 04 2024 08:58:00

%S 0,1,1,3,13,64,470,4921,83997,2411453,123544541,11537642646,

%T 2013389528672

%N Number of connected nonhamiltonian graphs with n nodes.

%D J. P. Dolch, Names of Hamiltonian graphs, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 259-271.

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219.

%H Jan Goedgebeur, Barbara Meersman, Carol T. Zamfirescu, <a href="https://arxiv.org/abs/1812.05650">Graphs with few Hamiltonian Cycles</a>, arXiv:1812.05650 [math.CO], 2018-2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonhamiltonianGraph.html">Nonhamiltonian Graph</a>.

%F a(n) = A001349(n) - A003216(n).

%e a(10) = A001349(10) - A003216(10) = number of connected graphs on 10 unlabeled nodes - number of Hamiltonian graphs with 10 nodes = 11716571 - 9305118 = 2411453.

%e a(11) = A001349(11) - A003216(11) = number of connected graphs on 11 unlabeled nodes - number of Hamiltonian graphs with 11 nodes = 1006700565 - 883156024 = 123544541.

%Y Cf. A000088, A001349, A003216, A022564.

%Y Cf. A246446 (number of not-necessarily-connected simple nonhamiltonian graphs on n nodes).

%K nonn,more

%O 1,4

%A _Jonathan Vos Post_, Mar 07 2007

%E Terms a(1)-a(9) corrected by _Travis Hoppe_, Jun 01 2014

%E a(11) added by _Eric W. Weisstein_, Aug 26 2014

%E a(12) from formula by _Falk Hüffner_, Aug 13 2017

%E a(13) added by _Jan Goedgebeur_, May 07 2019