Upper bounds on the extremal number of the 4-cycle

Jie Ma; Tianchi Yang

Upper bounds on the extremal number of the 4-cycle

Combinatorics 2021-10-13 v3

Authors: Jie Ma , Tianchi Yang

Abstract

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$ -vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a disproof of a conjecture of Erd\H{o}s from 1970s which asserts that $f(n)=\frac12 n^{3/2}+\frac14 n+o(n)$ .

Keywords

extremal number 4-cycle upper bound Erdős conjecture graph theory forbidden subgraph asymptotic bound

Cite

@article{arxiv.2107.11601,
  title  = {Upper bounds on the extremal number of the 4-cycle},
  author = {Jie Ma and Tianchi Yang},
  journal= {arXiv preprint arXiv:2107.11601},
  year   = {2021}
}

Comments

11 pages, 1 figure

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