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

Xuqing Bai; Weichan Liu; Xiangxiang Nie; Xin Zhang

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

Combinatorics 2026-07-17 v1 Discrete Mathematics

Authors: Xuqing Bai , Weichan Liu , Xiangxiang Nie , Xin Zhang

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 $\frac{323}{108}n-6 \qquad \text{for every } n\ge 27.$

Keywords

planar Turán number cycle C8 extremal graph theory upper bound

Cite

@article{arxiv.2607.16103,
  title  = {An improved upper bound for the planar Turán number of $C_8$},
  author = {Xuqing Bai and Weichan Liu and Xiangxiang Nie and Xin Zhang},
  journal= {arXiv preprint arXiv:2607.16103},
  year   = {2026}
}

Comments

The main paper is 8 pages long, with a 16-page appendix