论文标题
平面Turán的6周期数
Planar Turán number of the 6-cycle
论文作者
论文摘要
令$ {\ rm ex} _ {\ mathcal {p}}（n，n，t，h）$表示$ n $ -n $ vertex平面图中不包含$ h $作为子段的$ n $ vertex planar graph中的$ t $的最大副本数。当$ t = k_2 $，$ {\ rm ex} _ {\ mathcal {p}}（n，n，t，h）$是研究良好的功能，planarturán的$ h $的planarturán数量，由$ {\ rm ex} _ {\ rm ex} _ {\ rmm ex} _ {\ mathcal {p} {p}}} $表示。极端平面图的主题是由Dowden（2016）发起的。他获得了两个$ {\ rm ex} _ {\ Mathcal {p}}}（n，c_4）$和$ {\ rm ex} _ {\ Mathcal {p}}（n，c_5）$的尖锐上限。后来，Y. Lan等。继续此主题，并证明了$ {\ rm ex} _ {\ Mathcal {p}}}（n，c_6）\ leq \ frac {18（n-2）} {7} $。在本文中，我们给出了一个尖锐的上限$ {\ rm ex} _ {\ Mathcal {p}}}（n，c_6）\ leq \ leq \ frac {5} {2} {2} n-7 $，对于所有$ n \ geq 18 $，可改善LAN的结果。我们还对$ {\ rm ex} _ {\ mathcal {p}}（n，c_k）$提出了一个猜想，以$ k \ geq 7 $。
Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar Turán number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm ex}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${\rm ex}_{\mathcal{P}}(n,C_6) \leq \frac{5}{2}n-7$, for all $n\geq 18$, which improves Lan's result. We also pose a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$.
