An improved upper bound for the planar Turán number of $C_8$

Xuqing Bai, Weichan Liu, Xiangxiang Nie, Xin Zhang

arXiv:2607.16103v1 Primary: math.CO Secondary: cs.DM

Abstract

We prove that every $n$-vertex simple planar graph with no copy of $C_8$ has at most \[ \frac{69}{25}(n-2) \] edges, for every $n\ge 8$. This improves the best known bound \[ \frac{323}{108}n-6 \qquad \text{for every } n\ge 27. \]

Disclosure

“2 69 e(G) ≤ (n − 2) = (n − 2), α 25 contradicting the choice of G. The theorem follows. Declaration on AI-assisted tools OpenAI Codex was used only as an assistive tool for implementing and checking computer programs used in the finite computations. Data and code availability The source codes, recorded outputs, machine-readable certificates, and repr”
PDF page 7

Classification: Computational experiments or data processing
Multiplier: 3
Verified: 2026-08-23T18:56:51.030142Z

Structural counts

Pages 24 pdf

Theorems 2 source

Lemmas 18 source

Propositions 3 source

Corollaries 0 source

Definitions 2 source

Displayed equations 52 source

Bibliography entries 10 source

Appendix pages 17 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.