Some exact results of the generalized Tur\'an numbers for paths

For graphs $H$ and $F$ with chromatic number $\chi(F)=k$, we call $H$ strictly $F$-Tur\'an-good (or $(H, F)$ strictly Tur\'an-good) if the Tur\'an graph $T_{k-1}(n)$ is the unique $F$-free graph on $n$ vertices containing the largest number of copies of $H$ when $n$ is large enough. Let $F$ be a graph with chromatic number $\chi(F)\geq 3$ and a color-critical edge and let $P_\ell$ be a path with $\ell$ vertices. Gerbner and Palmer (2020, arXiv:2006.03756) showed that $(P_3, F)$ is strictly Tur\'an good if $\chi(H)\ge 4$ and they conjectured that (a) this result is true when $\chi(F)=3$, and, moreover, (b) $(P_\ell, K_k)$ is Tur\'an-good for every pair of integers $\ell$ and $k$. In the present paper, we show that $(H, F)$ is strictly Tur\'an-good when $H$ is a bipartite graph with matching number $\nu(H)=\lfloor \frac{|V(H)|}{2}\rfloor$ and $\chi(F)= 3$, as a corollary, this result confirms the conjecture (a); we also prove that $(P_\ell, F)$ is strictly Tur\'an-good for $2\le\ell\leq 6$ and $\chi(F)\ge 4$, this also confirms the conjecture (b) for $2\le\ell\leq 6$ and $k\ge 4$.