Denote by $\delta^+(G)$ the minimal out degree in $G$, and by $\delta^-(G)$ the minimal in-degree.
In his comment, Saeed have commented on a different extension that seems stronger, except it requires the graph to be strongly connected.
The strong connectivity was proven redundant for the Ghouila-Houri theorem about 30 years after it was first published, and I was wondering if the same holds for the extension Saeed presented.
So the question is:
Who proved (can anyone find the reference) that $\delta^+(G)+\delta^-(G) \geq n$ implies $G$ is Hamiltonian, given that $G$ is strongly connected?
Is the strong connectivity redundant here as well, i.e. Does $\delta^+(G)+\delta^-(G) \geq n$ imply strong connectivity?
(Note that while the graph obviously has to be strongly connected for it to be Hamiltonian, I'm asking whether this condition is implied by the degree conditions).