  C. Thomassen in \cite{[11]} suggested (see also \cite{[2]}, J. C.Bermond, C.
Thomassen, Cycles in Digraphs - A survey, J. Graph Theory 5 (1981) 1-43,
Conjectures 1.6.7 and 1.6.8) the following conjectures :
  1. Every 3-strongly connected digraph of order $n$ and with minimum degree at
least $n+1$ is strongly Hamiltonian-connected.
  2. Let $D$ be a 4-strongly connected digraph of order $n$ such that the sum
of the degrees of any pair of non-adjacent vertices is at least $2n+1$. Then
$D$ is strongly Hamiltonian-connected.
  We disprove Conjecture 1 and prove two results which provide some support for
Conjecture 2. The main goal of this article is to present the detailed proofs
of these results (in English).
