  Given a family of graphs $G_1,\dots,G_{n}$ on the same vertex set $[n]$, a
rainbow Hamilton cycle is a Hamilton cycle on $[n]$ such that each $G_i$
contributes exactly one edge. We prove that if $G_1,\dots,G_{n}$ are
independent samples of $G(n,p)$ on the same vertex set $[n]$, then for each
$\varepsilon>0$, whp, every collection of spanning subgraphs $H_i\subseteq
G_i$, with $\delta(H_i)\geq(\frac{1}{2}+\varepsilon)np$, admits a rainbow
Hamilton cycle. A similar result is proved for rainbow perfect matchings in a
family of $n/2$ graphs on the same vertex set $[n]$.
  Our method is likely to be applicable to further problems in the rainbow
setting, in particular, we illustrate how it works for finding a rainbow
perfect matching in the $k$-partite $k$-uniform hypergraph setting.
