Planar Tur\'an Number of the $\Theta_6$

TL;DR

This paper improves the upper bound on the maximum number of edges in large planar graphs that do not contain a specific Theta graph, and constructs examples that achieve this bound, advancing understanding of planar Turán numbers.

Contribution

The authors refine the upper bound for the planar Turán number of Theta_6 graphs and prove the bound is sharp by constructing extremal graphs.

Findings

Improved upper bound: /7 n - 48/7 for _ ext{p}(n, \u03b8_6)

Existence of infinitely many extremal graphs attaining the bound

Verification that the bound is sharp for large n

Abstract

Let $\mathcal{F}$ be a nonempty family of graphs. A graph $G$ is called $\mathcal{F}$-\textit{free} if it contains no graph from $\mathcal{F}$ as a subgraph. For a positive integer $n$, the \emph{planar Tur\'an number} of $\F$, denoted by $\ex_{\p}(n,\F)$, is the maximum number of edges in an $n$-vertex $\F$-free planar graph. Let $\Theta_k$ be the family of Theta graphs on $k\geq 4$ vertices, that is, graphs obtained by joining a pair of non-consecutive vertices of a $k$-cycle with an edge. Lan, Shi and Song determined an upper bound $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{36}{7}$, but for large $n$, they did not verify that the bound is sharp. In this paper, we improve their bound by proving $\text{ex}_{\mathcal{P}}(n,\Theta_6)\leq \frac{18}{7}n-\frac{48}{7}$ and then we demonstrate the existence of infinitely many positive integer $n$ and an $n$-vertex $\Theta_6$-free planar graph attaining the bound.