Generalized Planar Turán Numbers

Ervin Győri,Nika Salia,Addisu Paulos,Casey Tompkins,Oscar Zamora

doi:10.37236/9603

Ervin Győri, Nika Salia + Show 3 more

Open Access

https://doi.org/10.37236/9603

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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.

Highlights

For a given family of graphs F, we say that a graph is F-free if it contains no F ∈ F as a subgraph
In the case that F = {F }, we denote the generalized extremal function ex(n, H, F) by ex(n, H, F ) and we refer to F-free graphs as F -free
As a corollary of Theorem 5 we obtain the order of magnitude of the maximum number of cycles

Summary

Introduction

For a given family of graphs F, we say that a graph is F-free if it contains no F ∈ F as a subgraph. In the case that F = {F }, we denote the generalized extremal function ex(n, H, F) by ex(n, H, F ) and we refer to F-free graphs as F -free Problems of this type have a long history beginning with a result of Zykov [38] (and later independently Erdos [11]) who determined the value of ex(n, Kr, Kt) for any pair of cliques. In this paper we will consider a common generalization of the types of problems mentioned above To this end, let exP(n, H, F) denote the maximum number of copies of H possible in an n-vertex planar graph containing no graph F ∈ F as a subgraph (we write exP(n, H, F ) in the case F = {F }). The problems considered in the preceding paragraphs correspond to the special cases of exP(n, K2, F ) and exP(n, H, ∅), for given graphs F and H

Notation and Results

General Upper Bounds for Degenerate Graph Classes

The Number of Trees in Planar Graphs

The number of C5’s in C4-free Planar Graphs

The Number of Trees in Planar Graph with no even cycle of length at most 2