  Consider the random subgraph process on a base graph $G$ on $n$ vertices: a
sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$
obtained by choosing an ordering of the edges of $G$ uniformly at random, and
by sequentially adding edges to $G_0$, the empty graph on the vertex set of
$G$, according to the chosen ordering. We show that if $G$ has one of the
following properties:
  1. There is a positive constant $\varepsilon > 0$ such that $\delta (G) \geq
\left( \frac{1}{2} + \varepsilon \right) n$;
  2. There are some constants $\alpha, \beta >0$ such that every two disjoint
subsets $U,W$ of size at least $\alpha n$ have at least $\beta |U||W|$ edges
between them, and the minimum degree of $G$ is at least $(2\alpha + \beta
)\cdot n$; or:
  3. $G$ is an $(n,d,\lambda )$--graph, with $d\geq \frac{C\cdot n\cdot \log
\log n}{\log n}$ and $\lambda \leq \frac{c\cdot d^2}{n}$ for some absolute
constants $c,C>0$.
  then for a positive integer constant $k$ with high probability the hitting
time of the property of containing $k$ edge disjoint Hamilton cycles is equal
to the hitting time of having minimum degree at least $2k$. These results
extend prior results by by Johansson and by Frieze and Krivelevich, and answer
a question posed by Frieze.
