论文标题
$ k_1p _ {\ ell} \ cup k_2s _ {\ ell-1} $的Turán数字的一些结果
Some results on the Turán number of $k_1P_{\ell}\cup k_2S_{\ell-1}$
论文作者
论文摘要
图$ h $的Turán数字，用$ ex（n，h）$表示，是$ n $顶点上包含no $ h $的任何图中的最大边数。令$ p _ {\ ell} $表示$ \ ell $ vertices上的路径，$ s _ {\ ell-1} $在$ \ ell $ vertices和$ k_1p _ {\ ell} \ cup k_2s _ { $ p _ {\ ell} $和$ k_2 $ copies of $ s _ {\ ell-1} $。在2013年，Lidický等人。首先考虑了$ k_1p_4 \ cup k_2s_3 $的turán数字，以实现足够大的$ n $。在2022年，张和王提出了关于$ k_1p_ {2 \ ell} \ cup k_2s_ {2 \ ell-1} $的turán数量的猜想。在本文中，我们确定$ p _ {\ ell} \ cup ks _ {\ ell-1} $，$ k_1p_ {2 \ ell} \ cup k_2s_ {2 \ ell-1} $，$ 2p_5 \ cup ks_4 $ for $ n $ zhang，zhung as zhung yimpe zhang，相应的极端图也被完全表征。
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 $k_1P_{\ell}\cup k_2S_{\ell-1}$ 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 $k_1P_4\cup k_2S_3$ for sufficiently large $n$. In 2022, Zhang and Wang raised a conjecture about the Turán number of $k_1P_{2\ell}\cup k_2S_{2\ell-1}$. In this paper, we determine the Turán numbers of $P_{\ell}\cup kS_{\ell-1}$, $k_1P_{2\ell}\cup k_2S_{2\ell-1}$, $2P_5\cup kS_4$ for $n$ appropriately large, which implies the conjecture of Zhang and Wang. The corresponding extremal graphs are also completely characterized.
