On the number of Hamilton cycles in sparse random graphs

TL;DR

This paper establishes the asymptotic number of Hamilton cycles in sparse random graphs G(n,p) and shows that the first edge to achieve minimum degree 2 in the process creates a large number of Hamilton cycles.

Contribution

It provides the first precise asymptotic count of Hamilton cycles in sparse G(n,p) and extends to a hitting-time result in the random graph process.

Findings

Number of Hamilton cycles in G(n,p) is approximately n!p^n

In the random graph process, the initial edge creating minimum degree 2 yields about (ln n/e)^n Hamilton cycles

Results hold with high probability for p above the threshold (ln n + ln ln n + ω(1))/n

Abstract

We prove that the number of Hamilton cycles in the random graph G(n,p) is n!p^n(1+o(1))^n a.a.s., provided that p\geq (ln n+ln ln n+\omega(1))/n. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates (ln n/e)^n(1+o(1))^n Hamilton cycles a.a.s.