Spanning cycles in random directed graphs

TL;DR

This paper proves that in random directed graphs, all possible oriented cycles appear before the Hamilton cycle, and it determines the exact threshold for any specific cycle's appearance, confirming a conjecture.

Contribution

It establishes the order of appearance of all oriented cycles relative to the Hamilton cycle and finds precise thresholds for individual cycle appearances in random directed graphs.

Findings

All oriented cycles appear before the Hamilton cycle in random directed graphs.

The sharp threshold for the appearance of any specific oriented cycle is determined.

Results confirm a conjecture by Ferber and Long.

Abstract

We show that, in almost every $n$-vertex random directed graph process, a copy of every possible $n$-vertex oriented cycle will appear strictly before a directed Hamilton cycle does, except of course for the directed cycle itself. Furthermore, given an arbitrary $n$-vertex oriented cycle, we determine the sharp threshold for its appearance in the binomial random directed graph. These results confirm, in a strong form, a conjecture of Ferber and Long.