Some results on the Turán number of $k_1{P}_{\ell }\cup k_2{S}_{\ell -1}$

Abstract

The Turán number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices containing no $H$ as a subgraph. Let $P_{\ell }$ denote the path on $\ell$ vertices, $S_{\ell -1}$ denote the star on $\ell$ vertices and $k_1{P}_{\ell }\cup k_2{S}_{\ell -1}$ denote the path-star forest with disjoint union of $k_1$ copies of $P_{\ell }$ and $k_2$ copies of $S_{\ell -1}$. In 2013, Lidický et al. first considered the Turán number of $k_1{P}_4\cup k_2{S}_3$ for sufficiently large $n$. In 2022, Zhang and Wang raised a conjecture about the Turán number of $k_1{P}_{2\ell }\cup k_2{S}_{2\ell -1}$. In this paper, we determine the Turán numbers of $P_{\ell }\cup k{S}_{\ell -1}$, $k_1{P}_{2\ell }\cup k_2{S}_{2\ell -1}$, $2P_5\cup k{S}_4$ for $n$ appropriately large, which implies the conjecture of Zhang and Wang. The corresponding extremal graphs are also completely characterized.

13 pages