Power of k choices and rainbow spanning trees in random graphs

We consider the Erdos-Renyi random graph process, which is a stochastic process that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly at random from the set of missing edges. Let g(n, m) be a graph with in edges obtained after in steps of this process. Each edge e(i) (i = 1, 2,..., m) of g(n, m) independently chooses precisely k is an element of N colours, uniformly at random, from a given set of n - 1 colours (one may view e(i) as a multi-edge). We stop the process prematurely at time M when the following two events hold: g(n, M) is connected and every colour occurs at least once (M = ((n)(2)) if some colour does not occur before all edges are present; however, this does not happen asymptotically almost surely). The question addressed in this paper is whether g (n, M) has a rainbow spanning tree (that is, multicoloured tree on n vertices). Clearly, both properties are necessary for the desired tree to exist. In 1994, Frieze and McKay investigated the case k = 1 and the answer to this question is "yes" (asymptotically almost surely). However, since the sharp threshold for connectivity is n/2 log n and the sharp threshold for seeing all the colours is n/k log n, the case k = 2 is of special importance as in this case the two processes keep up with one another. In this paper, we show that asymptotically almost surely the answer is "yes" also for k >= 2.