On the threshold for rainbow connection number r in random graphs

TL;DR

This paper studies the threshold in Erdős-Rényi random graphs for the property that the rainbow connection number is at most a fixed integer r, providing bounds and conjectures for different values of r.

Contribution

The paper proposes an upper bound for the threshold of rainbow connection number r in random graphs and discusses differences between r=2 and r>=3 cases.

Findings

For r=2, threshold coincides with diameter 2 in random graphs.

Proposes an alternative threshold for r>=3.

Provides an upper bound for the threshold of rc(G) <= r.

Abstract

We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, i.e., a path where no two edges have the same colour. The minimum number of colours required for a rainbow colouring of the edges of G is called the rainbow connection number (or rainbow connectivity) rc(G) of G. We investigate sharp thresholds in the Erd\H{o}s-R\'enyi random graph for the property "rc(G) <= r" where r is a fixed integer. It is known that for r=2, rainbow connection number 2 and diameter 2 happen essentially at the same time in random graphs. For r >= 3, we conjecture that this is not the case, propose an alternative threshold, and prove that this is an upper bound for the threshold for rainbow connection number r.