Notes on sufficient conditions for a graph to be hamiltonian (Q1187097)

The authors prove the following lemma: If \(D\) is a weakly connected digraph of order \(p\), and if for each vertex \(v\), \(id(v)\geq p/2\) and \(od(v)\geq p/2\), then \(D\) is strongly connected. It is easy to see that the requirement of being weakly connected in the hypothesis of the lemma is redundant. In fact, in some textbooks on graph theory, the Ghouila- Houri theorem has already been stated as follows: If \(D\) is a simple digraph of order \(p\geq 3\) and if for each vertex \(v\), \(id(v)\geq p/2\) and \(od(v)\geq p/2\), then \(D\) is a hamiltonian digraph. See, for example, \textit{J. A. Bondy} and \textit{U. S. R. Murty} [Graph theory with applications. New York, NY: American Elsevier Publishing Co. (1976; Zbl 1226.05083), p. 178, Theorem 10.4] and \textit{F. Harary} [Graph Theory. Reading, Mass. etc.: Addison-Wesley (1969; Zbl 0182.57702) p. 209, Exercises 16.12].