Some exact results for non-degenerate generalized Turán problems

The generalized Turán number $\mathrm{e}\mathrm{x}(n,H,F)$ is the maximum number of copies of $H$ in $n$-vertex $F$-free graphs. We consider the case where $\chi (H)<\chi (F)$. There are several exact results on $\mathrm{e}\mathrm{x}(n,H,F)$ when the extremal graph is a complete $(\chi (F)-1)$-partite graph. We obtain multiple exact results with other kinds of extremal graphs.