Geodesic
Planar Turán numbers of cycles: a counterexample
Daniel W. Cranston1 ; Bernard Lidický2 ; Xiaonan Liu3 ; Abhinav Shantanam4
1Virginia Commonwealth University
2Department of Mathematics, Iowa State University, Ames, IA, USA;
3School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
4Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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The planar Turán 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 each $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known that hold with equality infinitely often. Ghosh, Győri, Martin, Paulos, 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.
DOI : 10.37236/10774
Classification : 05C30, 05C38, 05C35, 05C10
Mots-clés : planar Turán number, extremal planar graph

Daniel W. Cranston  1   ; Bernard Lidický  2   ; Xiaonan Liu  3   ; Abhinav Shantanam  4

1 Virginia Commonwealth University
2 Department of Mathematics, Iowa State University, Ames, IA, USA;
3 School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
4 Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada 
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     title = {Planar {Tur\'an} numbers of cycles: a counterexample},
     journal = {The electronic journal of combinatorics},
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Daniel W. Cranston; Bernard Lidický;  Xiaonan Liu;  Abhinav Shantanam. Planar Turán numbers of cycles: a counterexample. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10774

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