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The first part of this paper deals with an extension of Dirac's Theorem to directed graphs. It is related to a result often referred to as the Ghouila-Houri Theorem. Here we show that the requirement of being strongly connected in the hypothesis of the Ghouila-Houri Theorem is redundant.
The Second part of the paper shows that a condition on the number of edges for a graph to be hamiltonian implies Ore's condition on the degrees of the vertices.
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Michael Joseph Paul, Carmen Baytan Shershin, and Anthony Connors Shershin, “Notes on sufficient conditions for a graph to be Hamiltonian,” International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 4, pp. 825-827, 1991. doi:10.1155/S0161171291001138
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