Generalized Planar Turán Numbers
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Published:2021-11-19
Issue:4
Volume:28
Page:
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ISSN:1077-8926
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Container-title:The Electronic Journal of Combinatorics
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language:
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Short-container-title:Electron. J. Combin.
Author:
Győri Ervin,Paulos Addisu,Salia Nika,Tompkins Casey,Zamora Oscar
Abstract
In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
4 articles.
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1. Planar Graphs with the Maximum Number of Induced 6-Cycles;The Electronic Journal of Combinatorics;2023-10-06
2. The maximum number of copies of an even cycle in a planar graph;Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications;2023
3. The maximum number of paths of length three in a planar graph;Journal of Graph Theory;2022-04-19
4. Subgraph densities in a surface;Combinatorics, Probability and Computing;2022-01-11