Few $T$ copies in $H$-saturated graphs

A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $$\mathrm{s}\mathrm{a}\mathrm{t}$(n, H, F)$whichdenotestheminimumnumberofcopiesof$H$thatan$F$-saturated graph on $n$ vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelman’s generalized Tur'an problem, and letting $H = K_2$ recovers the well-studied saturation function. We provide a first investigation into this general function focusing on the cases where the host graph is either $K_s$ or $C_k$-saturated. Some representative interesting behavior is: (a) For any natural number $m$, there are graphs $H$ and $F$ such that $\mathrm{s}\mathrm{a}\mathrm{t}(n,H,F)=\mathrm{\Theta }(n^m)$. (b) For many pairs $k$ and $l$, we show $\mathrm{s}\mathrm{a}\mathrm{t}(n,C_l,C_k)=0$. In particular, we prove that there exists a triangle-free $C_k$-saturated graph on $n$ vertices for any $k>4$ and large enough $n$. © $\mathrm{s}\mathrm{a}\mathrm{t}(n,K_3,K_4)=n-2$, $$\mathrm{s}\mathrm{a}\mathrm{t}$(n, C_4, K_4) \sim \frac{n^2}{2}$,and$$\mathrm{s}\mathrm{a}\mathrm{t}$(n, C_6, K_5) \sim n^3$. We discuss several intriguing problems which remain unsolved.