Frieze–Krivelevich conjecture on Hamilton-cycle packings in random graphs

Let $p=p(n)$ satisfy $0\leq p(n)\leq 1$, and let $G\sim G(n,p)$ be the binomial random graph. Write $\delta(G)$ for its minimum degree. An edge-disjoint Hamilton-cycle packing is a collection of Hamilton cycles no two of which share an edge.

Frieze–Krivelevich conjecture. Asymptotically almost surely, $G$ contains

$$\left\lfloor\frac{\delta(G)}{2}\right\rfloor$$

edge-disjoint Hamilton cycles.

Each Hamilton cycle uses two edges at every vertex, so this is the natural trivial upper bound for such a packing. The conjecture was stated explicitly by Frieze and Krivelevich; the source proves it in the sparse range $\log n/n\leq p\leq n^{-1+\varepsilon}$ for some positive constant $\varepsilon$, while the full range of $p$ remains open in the source.