Counting Hamilton cycles in sparse random directed graphs

TL;DR

This paper strengthens known results by precisely estimating the typical number of directed Hamilton cycles in sparse random directed graphs, showing it is approximately n! times p^n under certain conditions.

Contribution

It provides a new asymptotic estimate for the number of Hamilton cycles in D(n,p) matching the threshold for Hamiltonicity, and extends to a hitting-time result in the random graph process.

Findings

Number of Hamilton cycles is typically n!(p(1+o(1)))^n in D(n,p).

At the hitting time when minimum degrees reach 1, there are about n!(log n / n)^n Hamilton cycles.

The results strengthen the understanding of Hamiltonicity thresholds and cycle counts in sparse random directed graphs.

Abstract

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles.