Extremal graphs without long paths and a given graph (2312.00620v1)
Abstract: 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.