Hamilton Cycles in a Class of Random Directed Graphs

AbstractWe prove that almost every 3-in, 3-out digraph is Hamiltonian

Hamilton cycles in random digraphs with minimum degree at least one

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} \

Hamilton cycles in a class of random directed graphs

Abstract: "We prove that almost every 5-in, 5-out digraph is Hamiltonian.