The Tur\'an number of $k_1P_{2l}\cup k_2S_{2l-1}$

Abstract

The Tur\'an 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 $P_k$ denote the path on $k$ vertices, $S_k$ denote the star on $k+1$ vertices and $k_1P_{2l}\cup k_2S_{2l-1}$ denote the path-star forest with disjoint union of $k_1$ copies of $P_{2l}s$ and $k_2$ copies of $S_{2l-1}s$. In 2019, Lan et al. determined the Tur\'an numbers of $kS_l$ and $k_1P_4\cup k_2S_3$. In 2022, Zhang et al. determined the Tur\'an numbers of $k_1P_6\cup k_2S_5$ and raised a conjecture of the Tur\'an numbers of $k_1P_{2l}\cup k_2S_{2l-1}$, where $k_1\geqslant 1$ and $l\geqslant 2$. In this paper, we study the hypothesis and determine the Tur\'an numbers of $k_1P_{2l}\cup k_2S_{2l-1}$ when $n$ is sufficiently large.