Geodesic
Construction of maximally non-Hamiltonian graphs
Diskretnaya Matematika, Tome 15 (2003) no. 2, pp. 89-102

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In this paper we study the maximally non-Hamiltonian graphs (MNH graphs), that is, the graphs whose property to be non-Hamiltonian ceases to exist as soon as an edge is added. We give rules how to construct MNH graphs which make it possible, starting from MNH graphs with some properties, to get MNH graphs of greater order. 
@article{DM_2003_15_2_a6,
     author = {P. V. Roldugin},
     title = {Construction of maximally {non-Hamiltonian} graphs},
     journal = {Diskretnaya Matematika},
     pages = {89--102},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2003_15_2_a6/}
}
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P. V. Roldugin. Construction of maximally non-Hamiltonian graphs. Diskretnaya Matematika, Tome 15 (2003) no. 2, pp. 89-102. http://geodesic.mathdoc.fr/item/DM_2003_15_2_a6/