show/hide this revision's text 2 added 5 characters in body

Here's an attempt to translate your problem into a language that is more intuitive for graph theorists:

Let $\Gamma$ be a directed random graph (with loops) of order $n$, where each arc is chosen independently with a fixed probability $p$. Let $\Delta$ be the line graph of $\Gamma$. Let $x$ be the maximal length of a trail in $\Gamma$ (a path in the line graph $\Delta$. \Delta$ of $\Gamma$).

  • How is $x$ distributed?
  • Under what conditions does $\Delta$ contain a Hamiltonian path?
show/hide this revision's text 1

Here's an attempt to translate your problem into a language that is more intuitive for graph theorists:

Let $\Gamma$ be a directed random graph (with loops) of order $n$, where each arc is chosen independently with a fixed probability $p$. Let $\Delta$ be the line graph of $\Gamma$. Let $x$ be the maximal length of a path in $\Delta$.

  • How is $x$ distributed?
  • Under what conditions does $\Delta$ contain a Hamiltonian path?