Rainbow Hamiltonicity in uniformly coloured perturbed digraphs

Abstract: We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every $\delta \in (0,1)$ there exists $C = C(\delta) > 0$ such that the following holds. Let $D_0$ be an $n$-vertex digraph with minimum semidegree at least $\delta n$ and suppose that each edge of the union of $D_0$ with the random digraph $D(n, p)$ on the same vertex set gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, $D_0 \cup D(n, p)$ has a rainbow directed Hamilton cycle. This improves a result of Aigner-Horev and Hefetz (2021) who proved the same in the undirected setting when the edges are coloured uniformly in a set of $(1 + \varepsilon)n$ colours.