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$contain a Hamiltonian path? 1 Here's an attempt to translate your problem into a language that is more intuitive for graph theorists: Let$\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?