The emergence of a giant rainbow component

TL;DR

This paper analyzes the size of the largest rainbow tree in a randomly edge-colored Erdős-Rényi graph, identifying phase transitions and asymptotic behaviors in different regimes, including near-critical and sparse cases.

Contribution

It determines the asymptotic order of the largest rainbow tree in weakly sub- and supercritical regimes and in the sparse regime, revealing phase transitions and the emergence of almost spanning rainbow structures.

Findings

Largest rainbow tree in near-critical regimes contains an almost spanning tree.

In the sparse regime, the largest rainbow tree has linear order.

For large enough parameters, the graph contains an almost spanning rainbow cycle.

Abstract

The random coloured graph $G_c(n,p)$ is obtained from the Erd\H{o}s-R\'{e}nyi 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 $c = \Theta(n)$, 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 \emph{weakly sub- and supercritical regimes}, when $p = \frac{1+\varepsilon}{n}$ for some $\varepsilon=\varepsilon(n)$ which satisfies $\varepsilon = o(1)$ and $|\varepsilon|^3 n\to\infty$. In particular, we show that in both of these regimes with high probability the largest component of $G_c(n,p)$ contains an almost spanning rainbow tree. We also consider the order of the largest rainbow tree in the \emph{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 an almost spanning rainbow cycle.