Power of k choices and rainbow spanning trees in random graphs

Abstract

We consider the Erdős-Rényi 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 m edges obtained after m steps of this process. Each edge ei (i = 1, 2,…, m) of G(n, m) independently chooses precisely k ∈ N colours, uniformly at random, from a given set of n–1 colours (one may view ei 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 = (n2) 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.