On the Number of Non-Hamiltonian Graphs

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.