On the threshold for rainbow connection number r in random graphs

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 Erdős–Rényi 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 .