Speaker: Ryan Martin
Title: Planar Tur\'an number of the 6-cycle.
Abstract: 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 Tur\'an 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. (2019) continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In this talk, 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.
Joint work with Debarun Ghosh, Ervin Gy\H{o}ri, Addisu Paulos, and Chuanqi Xiao.