On the threshold for rainbow connection number r in random graphs

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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.