The hitting time of rainbow connection number two

TL;DR

This paper investigates the rainbow connection number in random graphs, proving that near the diameter 2 threshold, the rainbow connection number is almost surely 2, and that the hitting times for diameter 2 and rainbow connection number 2 coincide.

Contribution

It establishes that in Erdős-Rényi random graphs near the diameter 2 threshold, the rainbow connection number is almost surely 2, and the hitting times for diameter 2 and rainbow connection number 2 are the same.

Findings

In $G(n,p)$ near the diameter 2 threshold, $rc(G)=2$ with high probability.

Hitting times for diameter 2 and rainbow connection number 2 coincide in the random graph process.

The result strengthens understanding of rainbow connectivity in sparse random graphs.

Abstract

In a graph $G$ with a given edge colouring, a rainbow path is a path all of whose edges have distinct colours. The minimum number of colours required to colour the edges of $G$ so that every pair of vertices is joined by at least one rainbow path is called the rainbow connection number $rc(G)$ of the graph $G$. For any graph $G$, $rc(G) \ge diam(G)$. We will show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ close to the diameter 2 threshold, with high probability if $diam(G)=2$ then $rc(G)=2$. In fact, further strengthening this result, we will show that in the random graph process, with high probability the hitting times of diameter 2 and of rainbow connection number 2 coincide.