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Planar Tur\'{a}n Numbers of Cycles: A Counterexample
- Source :
- Electronic Journal of Combinatorics. Vol. 29(3), 2022, #P3.31
- Publication Year :
- 2021
-
Abstract
- The planar Turan number $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ is the largest number of edges in an $n$-vertex planar graph with no $\ell$-cycle. For $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known that hold with equality infinitely often. Ghosh, Gy\"{o}ri, Martin, Paulo, and Xiao [arxiv:2004.14094] conjectured an upper bound on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ for every $\ell\ge 7$ and $n$ sufficiently large. We disprove this conjecture for every $\ell\ge 11$. We also propose two revised versions of the conjecture.<br />Comment: 9 pages, 2 figures
- Subjects :
- Mathematics - Combinatorics
05C35
Subjects
Details
- Database :
- arXiv
- Journal :
- Electronic Journal of Combinatorics. Vol. 29(3), 2022, #P3.31
- Publication Type :
- Report
- Accession number :
- edsarx.2110.02043
- Document Type :
- Working Paper