Dirac-type Problem of Rainbow matchings and Hamilton cycles in Random Graphs

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 $G_c$ 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 $H_c\subseteq G_c$, with $\delta(H_c)\geq(\frac{1}{2}+\varepsilon)np$, 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 $[n]$.