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 $D_{n,m}^{(\hat{I}´\ge 1)}$. We prove that if $m=\frac{n}{2}(\log n+2\log \log n+c_n)$ then \[ \lim_{n\to\infty}\Pr(D_{n,m}^{(δ\geq1)}\text{ is Hamiltonian})=\begin{cases}0&c_n\to-\infty.\\e^{-e^{-c}/4}&c_n\to c.\\1&c_n\to\infty.\end{cases} \]