Non-Isomorphic Smallest Maximally Non-Hamiltonian Graphs

A graph G is maximally non-hamiltonian (MNH) if G is not hamiltonian but becomes hamiltonian after adding an arbitrary new edge. Bondy 2] showed that the smallest size (= number of edges) in a MNH graph of order n is at least d 3n 2 e for n 7. The fact that equality may hold there for innnitely many n was suggested by Bollobbs 1]. This was connrmed by Clark, Entringer and Shapiro (see 5, 6]) and by Xiaohui, Wenzhou, Chengxue and Yuanscheng 8] who set the values of the size of smallest MNH graphs for all small remaining orders n. An interesting question of Clark and Entringer 5] is whether for innnitely many n the smallest MNH graph of order n is not unique. A positive answer-the existence of two non-isomorphic smallest MNH graphs for innnitely many n follows from results in 5], 4], 6] and 8]. But, there still exist innnitely many orders n for which only one smallest MNH graph of order n is known. We prove that for all n 88 there are at least (n) 3 smallest MNH graphs of order n, where limn!1 (n) = 1. Thus, there are only nitely many orders n for which the smallest MNH graph is unique.