[Paper Review] Extremal Theta-free planar graphs

Yongxin Lan, Yongtang Shi+1 more|arXiv (Cornell University)|Nov 5, 2017

Limits and Structures in Graph Theory11 references4 citations

TL;DR

This paper determines tight upper bounds for the maximum number of edges in $ heta_k$-free planar graphs, proving that $\text{ex}_{\mathcal{P}}(n,\Theta_4) \leq 12(n-2)/5$, $\text{ex}_{\mathcal{P}}(n,\Theta_5) \leq 5(n-2)/2$, and $\text{ex}_{\mathcal{P}}(n,\Theta_6) \leq 18(n-2)/7$, all of which are tight for infinitely many $n$. The results are established via face counting and Euler's formula, with equality achieved in specific extremal planar graphs composed of $K_5^-$ and 6-faces.

ABSTRACT

Given a family $\mathcal{F}$, a graph is $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. We study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230], that is, how many edges can an $\mathcal{F}$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,\mathcal{F})$ to be the maximum number of edges in an $\mathcal{F}$-free planar graph on $n $ vertices. Dowden obtained the tight bounds $ex_{_\mathcal{P}}(n,C_4)\leq15(n-2)/7$ for all $n\geq4$ and $ex_{_\mathcal{P}}(n,C_5)\leq(12n-33)/5$ for all $n\geq11$. In this paper, we continue to promote the idea of determining $ex_{_\mathcal{P}}(n,\mathcal{F})$ for certain classes $\mathcal{F}$. Let $Θ_k$ denote the family of Theta graphs on $k\ge4$ vertices, that is, graphs obtained from a cycle $C_k$ by adding an additional edge joining two non-consecutive vertices. The study of $ex_{_\mathcal{P}}(n,Θ_4)$ was suggested by Dowden. We show that $ex_{_\mathcal{P}}(n,Θ_4)\leq12(n-2)/5$ for all $n\geq 4$, $ex_{_\mathcal{P}}(n,Θ_5)\leq5(n-2)/2$ for all $n\ge5$, and then demonstrate that these bounds are tight, in the sense that there are infinitely many values of $n$ for which they are attained exactly. We also prove that $ex_{_\mathcal{P}}(n,C_6)\le ex_{_\mathcal{P}}(n,Θ_6)\le 18(n-2)/7$ for all $n\ge6$.

Motivation & Objective

• Address the open problem of determining extremal planar graphs that avoid Theta graphs $\Theta_k$ for $k \geq 4$, extending Dowden's foundational work on planar Turán numbers.
• Establish tight upper bounds on the maximum number of edges in $\mathcal{F}$-free planar graphs where $\mathcal{F}$ is a family of Theta graphs.
• Prove that the derived bounds are asymptotically tight by constructing infinite families of extremal graphs achieving equality.
• Explore connections between planar Turán numbers and planar anti-Ramsey numbers, particularly through the relationship $1 + \text{ex}_{\mathcal{P}}(n,\mathcal{H}) \leq \text{ar}_{\mathcal{P}}(n,H) \leq \text{ex}_{\mathcal{P}}(n,H)$ for $\mathcal{H} = \{H-e \mid e \in E(H)\}$.
• Provide structural characterizations of extremal graphs achieving the bounds, especially for $\Theta_6$-free graphs composed of $K_5^-$ and 6-faces.

Proposed method

• Apply Euler's formula $n - 2 \geq e(G) - f$ to relate the number of vertices, edges, and faces in a planar graph $G$.
• Use face counting techniques to bound the number of $i$-faces ($f_i$) by analyzing edge distributions across adjacent faces, particularly $E_{i,j}$, the set of edges shared by an $i$-face and a $j$-face.
• Establish inequalities using the constraints that $\Theta_k$-free graphs cannot contain certain face adjacency patterns, such as a 3-face adjacent to a 5-face in the case of $\Theta_6$-freeness.
• Utilize the fact that $K_5^-$-free graphs cannot contain a specific subgraph configuration (Figure 4b), which restricts the number of 3-faces and their adjacency, leading to tighter bounds.
• Construct a derived graph $G'$ by removing vertices from $K_5^-$ subgraphs in extremal graphs to analyze the structure of remaining faces and derive contradictions when equality is assumed.
• Combine face and edge counting with degree constraints, showing that minimum degree $\delta(G') = 2$ and that such graphs must contain a $\Theta_6$ subgraph, contradicting $\Theta_6$-freeness.

Experimental results

Research questions

• RQ1 What is the maximum number of edges in a $\Theta_4$-free planar graph on $n$ vertices, and is this bound tight?
• RQ2 How does the extremal function $\text{ex}_{\mathcal{P}}(n,\Theta_5)$ behave asymptotically, and for which $n$ is the bound achieved?
• RQ3 What is the tight upper bound for $\text{ex}_{\mathcal{P}}(n,\Theta_6)$, and can this bound be achieved for infinitely many $n$?
• RQ4 Can the bound $\text{ex}_{\mathcal{P}}(n,\Theta_6) \leq 18(n-2)/7$ be improved, and what structural properties prevent equality in $K_5^-$-free graphs?
• RQ5 Is $\text{ex}_{\mathcal{P}}(n,C_6)$ bounded above by $\text{ex}_{\mathcal{P}}(n,\Theta_6)$, and does equality hold for any $n$?

Key findings

• The maximum number of edges in a $\Theta_4$-free planar graph on $n$ vertices satisfies $\text{ex}_{\mathcal{P}}(n,\Theta_4) \leq 12(n-2)/5$ for all $n \geq 4$, and this bound is tight for infinitely many $n$.
• The bound $\text{ex}_{\mathcal{P}}(n,\Theta_5) \leq 5(n-2)/2$ holds for all $n \geq 5$, and equality is achieved for infinitely many $n$.
• For $\Theta_6$-free planar graphs, the bound $\text{ex}_{\mathcal{P}}(n,\Theta_6) \leq 18(n-2)/7$ holds for all $n \geq 6$, and this bound is tight for $n = 9$.
• Equality in $\text{ex}_{\mathcal{P}}(n,\Theta_6) \leq 18(n-2)/7$ is only possible if the extremal graph consists entirely of $K_5^-$ and 6-faces with no shared edges, but such graphs contain a $\Theta_6$ subgraph, so equality cannot be achieved.
• Since every $C_6$-free graph is $\Theta_6$-free, it follows that $\text{ex}_{\mathcal{P}}(n,C_6) \leq 18(n-2)/7$, with equality achieved at $n = 9$.
• An extremal $\Theta_6$-free graph achieving $18(n-2)/7$ edges would require a derived graph $G'$ with minimum degree 2 and alternating 3- and 6-faces, but such a graph must contain a $\Theta_6$ subgraph, contradicting $\Theta_6$-freeness.