Title: Rainbow Hamiltonicity in uniformly coloured perturbed graphs

Abstract: We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed graph. We show that for every $\delta \in (0,1)$ there exists $C = C(\delta) > 0$ such that the following holds. Let $G_0$ be an $n$-vertex graph with minimum degree at least $\delta n$ and suppose that each edge of the union of $G_0$, with the random graph $G(n, p)$ on the same vertex set, gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, $G_0 \cup G(n, p)$ has a rainbow Hamilton cycle. This improves a result of Aigner-Horev and Hefetz, who proved the same when the edges are coloured uniformly in a set of $(1 + \epsilon)n$ colours.