Hitting Time of Edge Disjoint Hamilton Cycles in Random Subgraph Processes on Dense Base Graphs

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