The Maximum Number of Paths of Length Four in a Planar Graph

Debarun Ghosh; Ervin Győri; Ryan R. Martin; Addisu Paulos; Nika Salia; Chuanqi Xiao; Oscar Zamora

The Maximum Number of Paths of Length Four in a Planar Graph

Combinatorics 2021-10-18 v3

Authors: Debarun Ghosh , Ervin Győri , Ryan R. Martin , Addisu Paulos , Nika Salia , Chuanqi Xiao , Oscar Zamora

Abstract

Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$ -vertex planar graph. The order of magnitude of $f(n,P_k)$ , where $P_k$ is a path on $k$ vertices, is $n^{{\lfloor{\frac{k-1}{2}}\rfloor}+1}$ . In this paper we determine the asymptotic value of $f(n,P_5)$ and give conjectures for longer paths.

Keywords

planar graphs paths maximum number of copies extremal graph theory asymptotic value

Cite

@article{arxiv.2004.09207,
  title  = {The Maximum Number of Paths of Length Four in a Planar Graph},
  author = {Debarun Ghosh and Ervin Győri and Ryan R. Martin and Addisu Paulos and Nika Salia and Chuanqi Xiao and Oscar Zamora},
  journal= {arXiv preprint arXiv:2004.09207},
  year   = {2021}
}

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9 pages, 2 figures

The Maximum Number of Paths of Length Four in a Planar Graph

Abstract

Keywords

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