论文标题
平面Turán循环数量的改进的下限
An improved lower bound for the planar Turán number of cycles
论文作者
论文摘要
图$ h $的平面图，由$ ex _ {_ \ mathcal {p}}}（n，h）$表示，是$ n $ dertices上平面图中最大数量的边缘，而不包含$ h $作为子分类。在本文中，我们继续研究Dowden发起的“极端”平面图的主题[J. Graph Demole 83（2016）213--230]。我们首先获得了$ ex _ {_ \ mathcal {p}}}（n，c_k）$的改进的下限，用于所有$ k \ ge 13 $和$ n \ ge 5（k-6+\ lfloor {（k-1）}/2 \ rfloor）每个$ k $和$ n $的结构提供了一个更简单的反例，对Ghosh，Győri，Martin，Paulos和Xiao [arxiv：2004.14094v1]的猜想最近被Cranston，Lidický，Liu，Liu和Shantanam [Electon] [Electon]拒绝了。 J. Combin。 29（3）（2022）\＃p3.31]每$ k \ ge 11 $和$ n $都足够大（作为$ k $的函数）。然后，我们证明$ ex _ {_ \ Mathcal {p}}（n，h^+）= ex _ {_ \ Mathcal {p}}}（n，h）$ for $ k \ ge 5 $和$ k \ ge 5 $ and $ n \ ge | ge | h | h | h | h | h |+++1 $将吊坠边缘添加到二维的顶点。
The planar Turán number of a graph $H$, denoted by $ex_{_\mathcal{P}}(n,H)$, is the largest number of edges in a planar graph on $n $ vertices without containing $H$ as a subgraph. In this paper, we continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230]. We first obtain an improved lower bound for $ex_{_\mathcal{P}}(n,C_k)$ for all $k\ge 13$ and $n\ge 5(k-6+\lfloor{(k-1)}/2\rfloor)(k-1)/2$; the construction for each $k$ and $n$ provides a simpler counterexample to a conjecture of Ghosh, Győri, Martin, Paulos and Xiao [arxiv:2004.14094v1], which has recently been disproved by Cranston, Lidický, Liu and Shantanam [Electron. J. Combin. 29(3) (2022) \#P3.31] for every $k\ge 11$ and $n$ sufficiently large (as a function of $k$). We then prove that $ex_{_\mathcal{P}}(n,H^+)=ex_{_\mathcal{P}}(n,H)$ for all $k\ge 5$ and $n\ge |H|+1$, where $H\in\{C_k, 2C_k\}$ and $H^+$ is obtained from $H$ by adding a pendant edge to a vertex of degree two.
