On the threshold for rainbow connection number $r$ in random graphs

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) ${\mathrm {rc}}(G)$ of $G$. We investigate sharp thresholds in the Erdős–Rényi random graph for the property “${\mathrm {rc}}(G)\le r$” 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 $r \ge 3$, 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 $r$.