On the size of maximally non-hamiltonian digraphs

A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open

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 infinitely many n was suggested by Bollobás [1]. This was confirmed 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 infinitely 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 infinitely many n follows from results in [5], [4], [6] and [8]. But, there still exist infinitely 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 ø(n) 3 smallest MNH..