  For a family of graphs $\mathcal{F}$, the Tur\'{a}n number
$ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph
containing no member of $\mathcal{F}$ as a subgraph. The maximum number of
edges in an $n$-vertex connected graph containing no member of $\mathcal{F}$ as
a subgraph is denoted by $ex_{conn}(n,\mathcal{F})$. Let $P_k$ be the path on
$k$ vertices and $H$ be a graph with chromatic number more than $2$. Katona and
Xiao [Extremal graphs without long paths and large cliques, European J.
Combin., 2023 103807] posed the following conjecture: Suppose that the
chromatic number of $H$ is more than $2$. Then
$ex\big(n,\{H,P_k\}\big)=n\max\big\{\big\lfloor
\frac{k}{2}\big\rfloor-1,\frac{ex(k-1,H)}{k-1}\big\}+O_k(1)$. In this paper, we
determine the exact value of $ex_{conn}\big(n,\{P_k,H\}\big)$ for sufficiently
large $n$. Moreover, we obtain asymptotical result for
$ex\big(n,\{P_k,H\}\big)$, which solves the conjecture proposed by Katona and
Xiao.
