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 ${\mathrm{sat}}_t(n,\mathfrak{R}(H))$, the minimum number of edges in a $t$-edge-coloured 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 colour from $\{1,2,\dots ,t\}$ creates a rainbow copy of $H$? Here, we completely characterize the growth rates of $\operatorname{sat}{t}\left(n, \mathfrak{R}{(H)}\right)$asafunctionof$n$,foranygraph$H$ belonging to a large class of connected graphs and for any $t\ge e(H)$. This classification includes all connected graphs of minimum degree $2$. In particular, we prove that $\operatorname{sat}{t}\left(n, \mathfrak{R}{(K_r)}\right)=\Theta(n\log n)$,forany$r\geq 3$and$t\geq {r \choose 2}$, thus resolving a conjecture of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new problems and conjectures.