A rainbow connectivity threshold for random graph families

TL;DR

This paper determines the threshold number of random graphs needed for a family to be rainbow connected with high probability, revealing a sharp phase transition based on the size of the family.

Contribution

It introduces a threshold for rainbow connectivity in random graph families and shows the threshold's concentration on at most three values, extending understanding of rainbow connectivity.

Findings

Rainbow connectivity occurs with high probability when the number of graphs exceeds the threshold.

Below the threshold, the family is unlikely to be rainbow connected.

The threshold is closely related to the diameter of the union of the graphs.

Abstract

Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in $\mathcal G$. We consider the case that $\mathcal G$ contains $s$ graphs, each sampled randomly from $G(n,p)$, with $n = |X|$ and $p = \frac{c \log n}{sn}$, where $c > 1$ is a constant. We show that when $s$ is sufficiently large, $\mathcal G$ is a.a.s. rainbow connected, and when $s$ is sufficiently small, $\mathcal G$ is a.a.s. not rainbow connected. We also calculate a threshold of $s$ for the rainbow connectivity of $\mathcal G$, and we show that this threshold is concentrated on at most three values, which are larger than the diameter of the union of $\mathcal G$ by about $\frac{\log n}{(\log \log n)^2}$. The same results also hold in a more traditional random rainbow setting, where we take a random graph $G\in G(n,p)$ with $p=\frac{c \log n}{n}$ ($c>1$) and color each edge of $G$ with a color chosen uniformly at random from the set $[s]$ of $s$ colors.