论文标题
普通平面图恩数字
Generalized Planar Turán Numbers
论文作者
论文摘要
在广义的Turán问题中，我们获得了图形$ h $和$ f $，并寻求最大化$ f $ f $ f $ f $ fod-free订单$ n $中的$ h $的副本数量。我们考虑了宿主图是平面的广义图拉问题。特别是，我们在平面图中获得了固定树的最大副本数量的数量级，其中包含均匀长度最多$ 2 \ ell $，对于所有$ \ ell $，$ \ ell $，$ \ ell \ geq 1 $。我们获得了平面$ C_4 $ Free图中给定长度的最大循环数量的数量级。在$ C_4 $ - Free Planar Graph中，给出了最大$ 5 $ CYCLE的确切结果。还引入了多个猜想。
In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $F$-free graph of order $n$. We consider generalized Turán problems where the host graph is planar. In particular we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geq 1$. We obtain the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.
