Rainbow hamilton cycles in random graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdős-Rényi random graph Gn,p is around p ∼ logn+log logn n . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3-uniform hypergraph by connecting 3-uniform hypergraphs to edge-colored graphs. In this work, we consider that setting of edge-colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of Gn,p are randomly colored from a set of (1 + o(1))n colors, with p = (1+o(1)) logn n , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).