The Hitting Time of Rainbow Connection Number Two

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) > diam(G). We will show that for the Erdős–Rényi 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.