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arXiv 1808.01487math.CO
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无极$H$的可平面图

Extremal $H$-free planar graphs

Yongxin Lan, Yongtang Shi, Zi-Xia Song

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摘要：

给定一个图$H$，如果它不包含$H$作为子图，则该图是$H$ -free的.我们继续研究“极值”可平面图的话题，即$n$顶点上的无$H$的平面图可以有多少条边？我们定义$ex_{_\mathcal{P}}(n,H)$为无$H$平面图中$n $顶点上的最大边数.我们首先得到了$H$上的几个充分条件，使得对所有$n\ge |V(H)|$都产生$ex_{_\mathcal{P}}(n,H)=3n-6$。我们发现，$H$的色数不起作用，在著名的Erdens-Stone定理。当$H$是一个轮子或一个星时，我们完全确定了$ex_{_\mathcal{P}}(n,H)$。最后，我们研究了$H$是$(t, r)$ -fan的情况，即$H$同构于$K_1+tK_{r-1}$，其中$t\ge2$和$r\ge 3$是整数。然而，当$H$是平面亚三次图时，确定$ex_{_\mathcal{P}}(n,H)$仍然是一个很大的问题。

英文摘要：

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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