On a rainbow version of Dirac's theorem

For a collection $\mathbf{G}=\{G_1,\dots ,G_s\}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $\mathbf{G}$-transversal if there exists a bijection $\varphi :E(H)\to [s]$ such that $e\in E(G_{\varphi (e)})$ for all $e\in E(H)$. We prove that for $\mathrm{\mid }V\mathrm{\mid }=s\ge 3$ and $\delta (G_i)\ge s\mathrm{/}2$ for each $i\in [s]$, there exists a $\mathbf{G}$-transversal that is a Hamilton cycle. This confirms a conjecture of Aharoni. We also prove an analogous result for perfect matchings.