[Paper Review] Extremal $H$-free planar graphs

Yongxin Lan, Yongtang Shi+1 more|arXiv (Cornell University)|Aug 4, 2018

Limits and Structures in Graph Theory8 references3 citations

TL;DR

This paper investigates extremal planar graphs that exclude a given graph $H$ as a subgraph, focusing on maximizing edges in $H$-free planar graphs on $n$ vertices. It establishes sufficient conditions under which the extremal function $\mathrm{ex}_{\mathcal{P}}(n,H) = 3n-6$ for all $n \geq |H|$, completely determines $\mathrm{ex}_{\mathcal{P}}(n,H)$ for wheels and stars, and provides tight bounds for $(t,r)$-fans, while leaving the case of planar subcubic graphs open.

ABSTRACT

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem. We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

Motivation & Objective

• To determine the maximum number of edges in an $H$-free planar graph on $n$ vertices, denoted $\mathrm{ex}_{\mathcal{P}}(n,H)$.
• To identify sufficient conditions on a graph $H$ such that $\mathrm{ex}_{\mathcal{P}}(n,H) = 3n - 6$ for all $n \geq |H|$.
• To completely determine $\mathrm{ex}_{\mathcal{P}}(n,H)$ when $H$ is a wheel or a star.
• To analyze the extremal function for $H$ being a $(t,r)$-fan, i.e., $K_1 + tK_{r-1}$ with $t \geq 2$, $r \geq 3$.
• To identify open problems, particularly for planar subcubic graphs.

Proposed method

• Use of Euler's formula and face counting to derive upper bounds on the number of edges in $H$-free planar graphs.
• Analysis of vertex degrees and face incidences, particularly $3$-faces, under $H$-freeness constraints.
• Application of structural graph theory to bound the number of $3$-faces incident to vertices of various degrees.
• Recursive construction of extremal graphs (e.g., $G_k$) to achieve tight lower bounds for specific $H$.
• Induction-based proof technique to establish upper bounds on $\mathrm{ex}_{\mathcal{P}}(n,K_1 + P_t)$ for $t \in \{4,5,6\}$.
• Use of the fact that $G$ is $(K_1 + 3K_2)$-free to limit the number of $3$-faces incident to high-degree vertices.

Experimental results

Research questions

• RQ1 Under what conditions on $H$ does $\mathrm{ex}_{\mathcal{P}}(n,H) = 3n - 6$ hold for all $n \geq |H|$?
• RQ2 What is the exact value of $\mathrm{ex}_{\mathcal{P}}(n,H)$ when $H$ is a wheel or a star?
• RQ3 What is the extremal function for $H = K_1 + tK_{r-1}$, i.e., a $(t,r)$-fan, with $t \geq 2$, $r \geq 3$?
• RQ4 Can tight bounds be established for $\mathrm{ex}_{\mathcal{P}}(n,H)$ when $H$ is a planar subcubic graph?
• RQ5 How does the chromatic number of $H$ influence $\mathrm{ex}_{\mathcal{P}}(n,H)$, given that it does not affect the extremal function in this setting?

Key findings

• For $H$-free planar graphs, $\mathrm{ex}_{\mathcal{P}}(n,H) = 3n - 6$ holds if $H$ is $K_4$-free and satisfies one of several degree or chromatic number conditions, such as $\chi(H) = 4$ and $n \geq |H| + 2$.
• The extremal function $\mathrm{ex}_{\mathcal{P}}(n,H)$ is completely determined for $H$ being a wheel or a star, with exact formulas derived.
• For $H = (t,r)$-fans with $t \geq 2$, $r \geq 3$, the paper provides tight upper and lower bounds, showing $\mathrm{ex}_{\mathcal{P}}(n,H) = \Theta(n)$.
• A construction of $G_k$ with $n = 24(k+1)$ vertices and $67n/24 - 4$ edges shows that the lower bound for $\mathrm{ex}_{\mathcal{P}}(n,K_1 + 3K_2)$ can be improved when $n$ is divisible by 24.
• An upper bound of $\mathrm{ex}_{\mathcal{P}}(n,K_1 + P_t) \leq \frac{13(t-1)n}{4t-2} - \frac{12(t-1)}{2t-1}$ is established for $t \in \{4,5,6\}$ via induction and face counting.
• The problem of determining $\mathrm{ex}_{\mathcal{P}}(n,H)$ for planar subcubic graphs $H$ remains open.