Generalized planar Turán numbers related to short cycles

Abstract

Given two graphs $H$ and $F$, the generalized planar Turán number ${\mathrm{ex}}_{\mathcal{P}}(n,H,F)$ is the maximum number of copies of $H$ that an $n$-vertex $F$-free planar graph can have. We investigate this function when $H$ and $F$ are short cycles. Namely, for large $n$, we find the exact value of ${\mathrm{ex}}_{\mathcal{P}}(n,C_l,C_3)$, where $C_l$ is a cycle of length $l$, for $4\le l\le 6$, and determine the extremal graphs in each case. Also, considering the converse of these problems, we determine sharp upper bounds for ${\mathrm{ex}}_{\mathcal{P}}(n,C_3,C_l)$, for $4\le l\le 6$.