Rainbow Hamilton cycles in randomly coloured randomly perturbed dense graphs

Given an $n$-vertex graph $G$ with minimum degree at least $dn$ for some fixed $d>0$, the distribution $G\cup \mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G\cup \mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r:=r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisfies $p := p(n) \geq C/n$,forsomefixed$C > 0$,and$r = (1 + o(1))n$. The number of colours used is clearly asymptotically best possible. In particular, this improves upon a recent result of Anastos and Frieze (2019) in this regard. As an intermediate result, which may be of independent interest, we prove that randomly edge-coloured sparse pseudo-random graphs a.a.s. admit an almost spanning rainbow path.