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arXiv · 2405.08162

Generalized planar Tur\'an numbers related to short cycles

Ervin Győri·Hilal Hama Karim
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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 $4\leq l\leq 6$.

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Ervin Győri, Hilal Hama Karim. 2024-05-13. Generalized planar Tur\'an numbers related to short cycles. https://arxiv.org/abs/2405.08162

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