How many randomly colored edges make a randomly colored dense graph rainbow Hamiltonian or rainbow connected?

摘要

In this paper, we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular, we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge-disjoint rainbow-Hamilton cycles w.h.p. We also ask when, in the resultant graph, every pair of vertices is connected by a rainbow path w.h.p.