Abstract
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
$C = C(\delta ) \gt 0$
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 a copy of the random digraph
$\mathbf{D}(n,C/n)$
on the same vertex set gets a colour in
$[n]$
independently and uniformly at random. Then, with high probability,
$D_0 \cup \mathbf{D}(n,C/n)$
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
$(1 + \varepsilon )n$
colours.
Publisher
Cambridge University Press (CUP)