On Hamiltonian Cycles in a 2-Strong Digraphs with Large Degrees and Cycles

In this note we prove: let D be a 2-strong digraph of order \(n\) such that its \(n - 1\) vertices have degrees at least \(n + k\) and the remaining vertex \(z\) has degree at least \(n - k - 4,\) where \(k\) is a nonnegative integer. If \(D\) contains a cycle of length at least \(n - k - 2\) passing through \(z,\) then \(D\) is Hamiltonian. This result is best possible in some sense.