The Turán number of $k_1{P}_{2l}\cup k_2{S}_{2l-1}$

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 $P_k$ denote the path on $k$ vertices, $S_k$ denote the star on $k+1$ vertices and $k_1{P}_{2l}\cup k_2{S}_{2l-1}$ 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 Turán numbers of $k{S}_l$ and $k_1{P}_4\cup k_2{S}_3$. In 2022, Zhang et al. determined the Turán numbers of $k_1{P}_6\cup k_2{S}_5$ and raised a conjecture of the Turán numbers of $k_1{P}_{2l}\cup k_2{S}_{2l-1}$, where $k_1⩾1$ and $l⩾2$. In this paper, we study the hypothesis and determine the Turán numbers of $k_1{P}_{2l}\cup k_2{S}_{2l-1}$ when $n$ is sufficiently large.

We need to continue to validate our results to ensure that they are completely correct