[Paper Review] Planar Tur\'an number of the 6-cycle

Debarun Ghosh, Ervin Győri+3 more|arXiv (Cornell University)|Apr 29, 2020

Limits and Structures in Graph Theory5 references4 citations

TL;DR

This paper establishes a sharp upper bound of $\frac{5}{2}n - 7$ on the planar Turán number $\mathrm{ex}_{\mathcal{P}}(n, C_6)$, improving upon the previous bound of $\frac{18(n-2)}{7}$. The authors prove this via structural analysis of $2$-connected $C_6$-free plane graphs with minimum degree at least 3, and construct extremal graphs achieving equality for infinitely many $n \equiv 2 \pmod{5}$, confirming tightness.

ABSTRACT

Let ${\ m ex}_{\\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\ m ex}_{\\mathcal{P}}(n,T,H)$ is the well studied function, the planar Tur\\'an number of $H$, denoted by ${\ m ex}_{\\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\ m ex}_{\\mathcal{P}}(n,C_4)$ and ${\ m ex}_{\\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\ m ex}_{\\mathcal{P}}(n,C_6)\\leq \\frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${\ m ex}_{\\mathcal{P}}(n,C_6) \\leq \\frac{5}{2}n-7$, for all $n\\geq 18$, which improves Lan's result. We also pose a conjecture on ${\ m ex}_{\\mathcal{P}}(n,C_k)$, for $k\\geq 7$.

Motivation & Objective

• To improve the known upper bound for the planar Turán number of the 6-cycle, $\mathrm{ex}_{\mathcal{P}}(n, C_6)$.
• To establish a tight bound that is achievable for infinitely many $n$ via explicit extremal graph constructions.
• To extend the framework of extremal planar graph theory beyond small cycles, particularly $C_4$, $C_5$, and $C_6$.
• To propose a general conjecture for the planar Turán number of longer cycles $C_\ell$ with $\ell \geq 7$.

Proposed method

• Prove Theorem 2: for $2$-connected, $C_6$-free plane graphs with $\delta(G) \geq 3$ and $n \geq 6$, $e(G) \leq \frac{5}{2}n - 7$ using block decomposition and degree-based inequalities.
• Use Euler's formula and face/vertex degree constraints to analyze a base graph $G_0$ with all faces of length 7 and vertices of degree 2 or 3.
• Construct an intermediate graph $G'$ by subdividing each edge of $G_0$ and adding chords between vertices at distance 2, preserving planarity and controlling cycle structure.
• Apply a discharging-like argument on blocks of the graph to bound $5v(G) - 2e(G)$, showing it is at least 14 for $n \geq 18$, implying $e(G) \leq \frac{5}{2}n - 7$.
• Construct extremal graphs achieving equality in the bound by using a family of graphs $G$ with $v(G) = \frac{18n+14}{5}$ and $e(G) = 9n$ for $n \equiv 2 \pmod{5}$.
• Generalize the construction to conjecture a formula for $\mathrm{ex}_{\mathcal{P}}(n, C_\ell)$ for $\ell \geq 7$ based on $\ell$-regular face structures and vertex degree control.

Experimental results

Research questions

• RQ1 What is the best possible upper bound for the number of edges in an $n$-vertex planar graph that avoids a $6$-cycle as a subgraph?
• RQ2 Can the bound $\mathrm{ex}_{\mathcal{P}}(n, C_6) \leq \frac{18(n-2)}{7}$ be improved, and if so, by how much?
• RQ3 For which values of $n$ does equality in the new bound $\frac{5}{2}n - 7$ hold, and can such extremal graphs be explicitly constructed?
• RQ4 Is there a general pattern or formula for the planar Turán number of longer cycles $C_\ell$ with $\ell \geq 7$?

Key findings

• The paper establishes a sharp upper bound: $\mathrm{ex}_{\mathcal{P}}(n, C_6) \leq \frac{5}{2}n - 7$ for all $n \geq 18$.
• This bound improves upon the previous best-known bound of $\frac{18(n-2)}{7}$, which was established by Lan et al.
• For $n \equiv 2 \pmod{5}$, there exist $C_6$-free planar graphs with $v(G) = \frac{18n+14}{5}$ and $e(G) = 9n$, achieving equality in the bound $\frac{5}{2}n - 7$.
• The extremal graphs are constructed by starting from a base graph $G_0$ with all faces of length 7 and vertices of degree 2 or 3, then applying edge subdivision and chord insertion.
• The proof relies on analyzing the block structure of the graph and showing that $5v(G) - 2e(G) \geq 14$ for $n \geq 18$, which implies the edge bound.
• The authors conjecture that for $\ell \geq 7$, $\mathrm{ex}_{\mathcal{P}}(n, C_\ell) \leq \frac{3(\ell-1)}{\ell}n - \frac{6(\ell+1)}{\ell}$ for sufficiently large $n$, based on a generalized construction.