Hamilton cycles in random digraphs with minimum degree at least one

Abstract: We study the existence of a directed Hamilton cycle in random digraphs with $m$ edges where we condition on minimum in- and out-degree at least one. Denote such a random graph by $D_{n,m}^{{(\delta\geq1)}$.} We prove that if $m=\tfrac n2(\log n+2\log\log n+c_n)$ then [ \lim_{n\to\infty}\Pr(D_{n,m}^{{(\delta\geq1)}\text{} is Hamiltonian})=\begin{cases}0&c_n\to-\infty.\e^{{-e{-c}/4}&c_n\to} c.\1&c_n\to\infty.\end{cases} ]