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

An infinite family of planar non-Hamiltonian bihomogeneously traceable oriented graphs

We answer an open question on planar non-hamiltonian bihomogeneously traceable digraphs without opposite arcs by constructing an infinite family of such graphs