Rainbow saturation of graphs

In this paper, we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph H and an integer t, what is satt(n,R(H)), the minimum number of edges in a t-edge-colored graph G on n vertices such that G does not contain a rainbow copy of H, but adding to G a new edge in any color from {1,2, horizontal ellipsis ,t} creates a rainbow copy of H? Here, we completely characterize the growth rates of satt(n,R(H)) as a function of n, for any graph H belonging to a large class of connected graphs and for any t >= e(H). This classification includes all connected graphs of minimum degree 2. In particular, we prove that satt(n,R(Kr))=Theta(nlogn), for any r >= 3 and t >= mml:mfenced close=) open=(r2, thus resolving a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new problems and conjectures.