Rainbow spanning trees in uniformly coloured perturbed graphs

Abstract

We consider the problem of finding a copy of a rainbow spanning bounded-degree tree in the uniformly edge-coloured randomly perturbed graph. Let $G_0$ be an $n$-vertex graph with minimum degree at least $\delta n$, and let $T$ be a tree on $n$ vertices with maximum degree at most $d$, where $\delta \in (0,1)$ and $d \ge 2$ are constants. We show that there exists $C = C(\delta, d) > 0$ such that, with high probability, if the edges of the union $G_0 \cup \gnp{n}{C/n}$ are uniformly coloured with colours in $[n-1]$, then there is a rainbow copy of $T$. Our result resolves in a strong form a conjecture of Aigner-Horev, Hefetz and Lahiri.