I am studying a paper named "Longest paths in digraphs".
In this paper they proved that for a connected digraph having the in-degree of any vertex at least $h$ and the out-degree of any vertex at least $k$ we can find a directed path having $\min(n,h+k+2)$ vertices.
But the paper was written in 1981, so I was searching for a newer and stronger result that gives a sufficient condition on the degrees of the vertices of a digraph to find a directed path of a given length. I have been searching for a while but didn't find any, so is it still the strongest result? Or what is the strongest result now?
There is also a conjecture by Thomassen that states
If a digraph $D$ has minimum indegree and outdegree at least $k$ and if any two vertices of $D$ are on a common circuit, then $D$ contains a circuit of length greater than or equal to $\min(2k,n)$.
Is this conjecture solved yet?