A Rainbow Connectivity Threshold for Random Graph Families

Given a family \(\mathcal G\) of graphs on common vertex set X, we say that is rainbow connected if for every pair \(u,v \in X\), there exists path from u to v uses at most one edge each graph G\). We consider the case contains s graphs, sampled randomly G(n, p), with \(n = |X|\) and \(p \frac{c \log n}{sn}\), where \(c > 1\) constant. show threshold three consecutive integer values such when greater than or equal this threshold, a.a.s. connected, below not connected.