Geodesic
On the Number of Non-Hamiltonian Graphs
Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 702-710

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In this paper, maximal non-Hamiltonian graphs (MNH graphs), i.e., non-Hamiltonian graphs such that the addition of any new edge violates their property of being non-Hamiltonian are studied. It is shown that the study of MNH graphs can be reduced to the study of the so-called simplified MNH graphs. Restrictions on the structure of maximal cliques of simplified MNH graphs are obtained, the orders and the number of such graphs are estimated. 
@article{MZM_2004_75_5_a6,
     author = {P. V. Roldugin},
     title = {On the {Number} of {Non-Hamiltonian} {Graphs}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {702--710},
     publisher = {mathdoc},
     volume = {75},
     number = {5},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_75_5_a6/}
}
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P. V. Roldugin. On the Number of Non-Hamiltonian Graphs. Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 702-710. http://geodesic.mathdoc.fr/item/MZM_2004_75_5_a6/