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On the size of maximally non-Hamiltonian digraphs. (English) Zbl 1416.05166
Summary: 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 M. Lewin [J. Comb. Theory, Ser. B 18, 175–179 (1975; Zbl 0297.05119)], but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open.
MSC:
05C45 Eulerian and Hamiltonian graphs
05C20 Directed graphs (digraphs), tournaments
05C12 Distance in graphs
05C35 Extremal problems in graph theory
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