The Planar Tur\'an Number of $\Theta_6$-graphs

TL;DR

This paper determines the maximum number of edges in large planar graphs that avoid specific 6-cycle graphs with diagonals, providing exact extremal constructions and sharp bounds.

Contribution

It introduces the first precise bounds for the planar Turán number of two particular $ heta_6$-graphs and constructs extremal graphs demonstrating these bounds.

Findings

Exact maximum edge counts for the two $ heta_6$-graphs

Infinitely many extremal constructions

Sharp bounds with minor additive error in one case

Abstract

There are two particular $\Theta_6$-graphs - the 6-cycle graphs with a diagonal. We find the planar Tur\'an number of each of them, i.e. the maximum number of edges in a planar graph $G$ of $n$ vertices not containing the given $\Theta_6$ as a subgraph and we find infinitely many extremal constructions showing the sharpness of these results - apart from a small additive constant error in one of the cases.