The emergence of a giant rainbow component

Abstract

The random coloured graph $G_c(n,p)$ is obtained from the Erdős–Rényi binomial random graph $G(n,p)$ by assigning to each edge a colour from a set of $c$ colours independently and uniformly at random. It is not hard to see that, when $c=\Theta \left(n\right)$, the order of the largest rainbow tree in this model undergoes a phase transition at the critical point $p=\frac{1}{n}$. In this paper, we determine the asymptotic order of the largest rainbow tree in the weakly sub- and supercritical regimes, when $p=\frac{1+\varepsilon }{n}$ for some $\varepsilon =\varepsilon \left(n\right)$ which satisfies $\varepsilon =o\left(1\right)$ and ${\left|\varepsilon \right|}^{3}n\to \infty$. In particular, we show that in both of these regimes with high probability the largest component of $G_c(n,p)$ contains a rainbow tree which is almost spanning. We also consider the order of the largest rainbow tree in the sparse regime, when $p=\frac{d}{n}$ for some constant $d>1$. Here we show that the largest rainbow tree has linear order, and, moreover, for $d$ and $c$ sufficiently large, with high probability $G_c(n,p)$ even contains a rainbow cycle which is almost spanning.