Generalized planar Tur\'an numbers related to short cycles

TL;DR

This paper determines the maximum number of certain short cycle subgraphs in large planar graphs that are free of other short cycles, providing exact values and extremal graph characterizations.

Contribution

It establishes exact values of generalized planar Turán numbers for cycles, specifically for $ ext{ex}_ ext{P}(n, C_l, C_3)$ with $4 \\leq l \\leq 6$, and bounds for the reverse cases.

Findings

Exact values for $ ext{ex}_ ext{P}(n, C_l, C_3)$ for $l=4,5,6$.

Characterization of extremal graphs achieving these values.

Sharp upper bounds for $ ext{ex}_ ext{P}(n, C_3, C_l)$ for $l=4,5,6$.

Abstract

Given two graphs $H$ and $F$, the generalized planar Tur\'an 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 $4\leq l\leq 6$, 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 $4\leq l\leq 6$.