Rainbow Hamiltonicity in uniformly coloured perturbed digraphs

We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every δ ∈ (0,1) there exists C = C(δ ) > 0 such that the following holds. Let D₀ be an n -vertex digraph with minimum semidegree at least δ n and suppose that each edge of the union of D₀ with a copy of the random digraph D(n,C/n) on the same vertex set gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, D₀ ∪ D(n,C/n) has a rainbow directed Hamilton cycle. This improves a result of Aigner-Horev and Hefetz ((2021) SIAM J. Discrete Math. 35 (3) 1569–1577), who proved the same in the undirected setting when the edges are coloured uniformly in a set of (1 + ε )n colours.