On Hamilton cycles in Erd\H{o}s-R\'{e}nyi subgraphs of large graphs

Given a graph $\Gamma = (V, E)$ on $n$ vertices and $m$ edges, we define the Erd\H{o}s-R\'{e}nyi graph process with host $\Gamma$ as follows. A permutation $e_1,\dots,e_m$ of $E$ is chosen uniformly at random, and for $t\leq m$ we let $\Gamma_t = (V, \{e_1,\dots,e_t\})$. Suppose the minimum degree of $\Gamma$ is $\delta(\Gamma) \geq (1/2 + \varepsilon)n$ for some constant $\varepsilon > 0$. Then with high probability, $\Gamma_t$ becomes Hamiltonian at the same moment that its minimum degree becomes at least two. Given $0\leq p\leq 1$ we let $\Gamma_p$ be the Erd\H{o}s-R\'{e}nyi subgraph of $\Gamma$, obtained by retaining each edge independently with probability $p$. When $\delta(\Gamma)\geq (1/2 + \varepsilon)n$, we provide a threshold function $p_0$ for Hamiltonicity, such that if $(p-p_0)n\to -\infty$ then $\Gamma_p$ is not Hamiltonian whp, and if $(p-p_0)n\to\infty$ then $\Gamma_p$ is Hamiltonian whp.