Rainbow Arborescence in Random Digraphs. Issue 3 (7th October 2015) - Bal, Deepak; Bennett, Patrick; Cooper, Colin; Frieze, Alan; Prałat, Paweł
Abstract: We consider the Erdős–Rényi random directed graph process, which is a stochastic process that starts with n vertices and no edges, and at each step adds one new directed edge chosen uniformly at random from the set of missing edges. Let D ( n, m ) be a graph with m edges obtained after m steps of this process. Each edge e i ( i = 1, 2, ..., m ) of D ( n, m ) independently chooses a color, taken uniformly at random from a given set of n ( 1 + O ( log log n / log n ) ) = n ( 1 + o ( 1 ) ) colors. We stop the process prematurely at time M when the following two events hold: D ( n, M ) has at most one vertex that has in‐degree zero and there are at least n − 1 distinct colors introduced ( M = n ( n − 1 ) if at the time when all edges are present there are still less than n − 1 colors introduced; however, this does not happen asymptotically almost surely). The question addressed in this article is whether D ( n, M ) has a rainbow arborescence (i.e. a directed, rooted tree on n vertices in which all edges point away from the root and all the edges are different colors). Clearly, both properties are necessary for the desired tree to exist and we show that, asymptotically almost surely, the answer to this question is "yes."